Combinatorial index formulas for Lie algebras of seaweed type

TL;DR

This paper develops combinatorial formulas to compute the index of seaweed subalgebras in Lie algebras of type D, extending previous types, and provides closed-form polynomial gcd formulas based on associated meander structures.

Contribution

It introduces comprehensive combinatorial formulas for the index of seaweed subalgebras in type D Lie algebras using meander tail analysis, extending classical types.

Findings

Formulas for the index based on connected components of meanders.

Closed-form polynomial gcd formulas for the index.

Unified combinatorial approach for classical types.

Abstract

Analogous to the types A, B, and C cases, we address the computation of the index of seaweed subalgebras in the type-D case. Formulas for the algebra's index can be computed by counting the connected components of its associated meander. We focus on a set of distinguished vertices of the meander, called the tail of the meander, and using the tail, we provide comprehensive combinatorial formulas for the index of a seaweed in all the classical types. Using these formulas, we provide all general closed-form index formulas where the index is given by a polynomial greatest common divisor formula in the sizes of the parts that define the seaweed.

Equations

$$\mathrm{ind}\mathfrak{g}=\underset{f\in {\mathfrak{g}}^{\ast }}{\min }\dim (\ker (B_f)),$$

$$B_f(x,y)=f([x,y])\text{ for}x,y\in \mathfrak{g},$$

$$\ker B_f=\{x\in \mathfrak{g}|f[x,y]=0\text{ for all}y\in \mathfrak{g}\}.$$

$$\gcd (f_1(a_1,\dots ,a_m),f_2(a_1,\dots ,a_m)).$$

$$\sum _{i,j}[F{]}_{ij}^{-1}{x}_i\wedge x_j$$

$$\mathfrak{g}={\mathfrak{u}}_{+}\oplus \mathfrak{h}\oplus {\mathfrak{u}}_{-},$$

$${\phi }_{\mathrm{A}}(\underline{a})=\{{\alpha }_{a_1},{\alpha }_{a_1+a_2},\dots ,{\alpha }_{a_1+a_2+\cdots +a_{m-1}}\}.$$

$${\mathfrak{p}}_n^{\mathrm{A}}(\underline{a}|\underline{b})={\mathfrak{p}}_n^{\mathrm{A}}({\phi }_{\mathrm{A}}(\underline{a})|{\phi }_{\mathrm{A}}(\underline{b})).$$

$$\gcd (a+b,b+c)-1.$$

$$\mathfrak{s}p(2n)=\left\{\left[\begin{array}{cc}A& B\\ C& -\hat{A}\end{array}\right]:B=\hat{B},C=\hat{C}\right\},$$

$${\phi }_{\mathrm{C}}(\underline{a})=\{{\alpha }_{a_1},{\alpha }_{a_1+a_2},\dots ,{\alpha }_{a_1+a_2+\cdots +a_m}\},$$

$${\mathfrak{p}}_n^{\mathrm{C}}(\underline{a}|\underline{b})={\mathfrak{p}}_n^{\mathrm{C}}({\phi }_{\mathrm{C}}(\underline{a})|{\phi }_{\mathrm{C}}(\underline{b})).$$

$$T=T_n(\underline{a}|\underline{b})=\left(T_n(\underline{a})\cup T_n(\underline{b})\right)\setminus \left(T_n(\underline{a})\cap T_n(\underline{b})\right).$$

$$\mathrm{ind}{\mathfrak{p}}_n^{\mathrm{C}}((a)|(b))=\{\begin{array}{ll}n-a+\lfloor \frac{a-b}{2}\rfloor ,& \text{ if }a-b\text{ is even;}\\ n-a+\lfloor \frac{a-b-1}{2}\rfloor ,& \text{ if }a-b\text{ is odd}.\end{array}$$

$$\mathrm{ind}{\mathfrak{p}}_n^{\mathrm{C}}((a,b)|(c))=\gcd (a+b,b+c)-1.$$

$$\mathfrak{s}o(2n)=\left\{\left[\begin{array}{cc}A& B\\ C& -\hat{A}\end{array}\right]:B=-\hat{B},C=-\hat{C}\right\},$$

$${\alpha }_1$$

$${\phi }_{\mathrm{D}}(\Psi )=(s_1,s_2-s_1,s_3-s_2,\dots ,s_j-s_{j-1}),$$

$${\mathfrak{p}}_n^{\mathrm{D}}(\Psi |{\Psi }^{\prime })={\mathfrak{p}}_n^{\mathrm{D}}(\underline{a}|\underline{b}),$$

$$T_n^{\mathrm{D}}(\underline{a}|\underline{b})=\{\begin{array}{ll}{T}_n^{\mathrm{C}}(\underline{a}|\underline{b}),& \text{ if }t\text{ is even;}\\ T_n^{\mathrm{C}}(\underline{a}|\underline{b})+v_{1+\sum a_i},& \text{ if }t\text{ is odd and }\sum a_i<n;\\ T_n^{\mathrm{C}}(\underline{a}|\underline{b})-v_n,& \text{ if }t\text{ is odd and }\sum a_i=n.\end{array}$$

$$\mathrm{ind}{\mathfrak{p}}_n^{\mathrm{D}}(\underline{a}|\underline{b})=a_1+\mathrm{ind}{\mathfrak{p}}_{n-a_1}^{\mathrm{D}}((a_2,a_3,\dots a_m)|(b_2,b_3,\dots b_r)).$$

$$\mathrm{ind}{\mathfrak{p}}_n^{\mathrm{D}}(\underline{a}|\underline{b})=\{\begin{array}{ll}\mathrm{ind}{\mathfrak{p}}_{n-a_1}^{\mathrm{D}}((a_2,a_3,\dots a_m)|(b_1-2a_1,a_1,b_2,b_3,\dots b_r)),& \text{ if }{a}_1\le b_1/2;\\ \mathrm{ind}{\mathfrak{p}}_{n-b_1+a_1}^{\mathrm{D}}((2a_1-b_1,a_2,a_3,\dots a_m)|(a_1,b_2,b_3,\dots b_r)),& \text{ if }{a}_1>b_1/2.\end{array}$$

$$\mathrm{ind}{\mathfrak{p}}_n^{\mathrm{D}}(\underline{a}|\varnothing )=\{\begin{array}{ll}\sum _{i=1}^m\lfloor \frac{a_i}{2}\rfloor +1,& \text{ if }{a}_m=1;\\ \sum _{i=1}^m\lfloor \frac{a_i}{2}\rfloor -1,& \text{ otherwise.}\end{array}$$

$$f\left(M_n^{\mathrm{D}}(\underline{a}|\varnothing )\right)=\{\begin{array}{ll}\sum _{i=1}^m\lfloor \frac{a_i}{2}\rfloor =\mathrm{ind}{\mathfrak{p}}_n^{\mathrm{D}}(\underline{a}|\varnothing ),& \text{ if }n\text{ is even;}\\ \sum _{i=1}^m\lfloor \frac{a_i}{2}\rfloor +1=\mathrm{ind}{\mathfrak{p}}_n^{\mathrm{D}}(\underline{a}|\varnothing ),& \text{ if }n\text{ is odd and }{a}_m=1;\\ \sum _{i=1}^m\lfloor \frac{a_i}{2}\rfloor -1=\mathrm{ind}{\mathfrak{p}}_n^{\mathrm{D}}(\underline{a}|\varnothing ),& \text{ if }n\text{ is odd and }{a}_m>1.\end{array}$$

$$f\left(M_n^{\mathrm{D}}(\underline{a}|\underline{b})\right)=f(H)+f(G)=a_1+\mathrm{ind}{\mathfrak{p}}_{n-a_1}^{\mathrm{D}}((a_2,a_3,\dots a_m)|(b_2,b_3,\dots b_t)).$$

$$G=M_{n-a_1}^{\mathrm{D}}((a_2,a_3,\dots a_m)|(b_1-2a_1,a_1,b_2,b_3,\dots b_t))$$

$$f\left(M_n^{\mathrm{D}}(\underline{a}|\underline{b})\right)=f(G)=\mathrm{ind}{\mathfrak{p}}_{n-a_1}^{\mathrm{D}}((a_2,a_3,\dots a_m)|(b_1-2a_1,a_1,b_2,b_3,\dots b_t)).$$

$$G=M_{n-b_1+a_1}^{\mathrm{D}}((2a_1-b_1,a_2,a_3,\dots a_m)|(a_1,b_2,b_3,\dots b_t))$$

$$f\left(M_n^{\mathrm{D}}(\underline{a}|\underline{b})\right)=f(G)=\mathrm{ind}{\mathfrak{p}}_{n-b_1+a_1}^{\mathrm{D}}((2a_1-b_1,a_2,a_3,\dots a_m)|(a_1,b_2,b_3,\dots b_t)).$$

$$\mathrm{ind}{\mathfrak{p}}_n^{\mathrm{D}}\frac{a}{b}=\{\begin{array}{ll}n-a+\lfloor \frac{a-b}{2}\rfloor ,& \text{ if }a\text{ is even};\\ n-a-1+\lfloor \frac{a-b-1}{2}\rfloor ,& \text{ if }a\text{ is odd}.\end{array}$$

Full text

∗*Department of Mathematics, Lehigh University, Bethlehem, PA, USA

∗∗FactSet Research Systems, New York, NY, USA

Mathematics Subject Classification 2010: 17B08

Key Words and Phrases: Frobenius Lie algebra, special orthogonal Lie algebra, seaweed, index, meander

1 Introduction

The index of a Lie algebra $\mathfrak{g}$ is an important algebraic invariant introduced by Dixmier ([7], 1974) and is defined by

[TABLE]

where $f$ is an element of the linear dual $\mathfrak{g}^{*}$, and $B_{f}$ is the associated skew-symmetric Kirillov form defined by

[TABLE]

and

[TABLE]

Here, we focus on combinatorial mechanisms to compute the index of certain subalgebras of the classical Lie algebras, which are the evocatively-named seaweed algebras. These algebras, along with their suggestive name, were first introduced by Dergachev and A. Kirillov in [6], where they defined such algebras as subalgebras of $\mathfrak{gl}(n)$ preserving certain flags of subspaces developed from two compositions of $n$. The evocative “seaweed” comes from the wavy shape the algebra demonstrates when exhibited in its standard matrix representation. The type-A case, $A_{n-1}=\mathfrak{sl}(n)$, is considered by requiring the elements of the seaweed in $\mathfrak{gl}(n)$ to have trace zero. Subsequently in [14], Panyushev extended the Lie-theoretic definition of seaweed algebras to the reductive case. If $\mathfrak{p}$ and $\mathfrak{p^{\prime}}$ are parabolic subalgebras of a reductive Lie algebra $\mathfrak{g}$ such that $\mathfrak{p}+\mathfrak{p^{\prime}}=\mathfrak{g}$, then $\mathfrak{p}\cap\mathfrak{p^{\prime}}$ is called a seaweed subalgebra of $\mathfrak{g}$ or simply $seaweed$ when $\mathfrak{g}$ is understood. As a result of this definition, Joseph has elsewhere [12] called seaweed algebras biparabolic. Joseph also showed, in response to a conjecture by Tauvel and Yu in [17], that the index of a seaweed is bounded by the algebra’s rank [12].

To facilitate the computation of the index of seaweed subalgebras of $\mathfrak{gl}(n)$, the authors in [6] introduced the notion of a meander – a planar graph representation of the seaweed algebra. The main result of [6] is that the index of a seaweed can be computed based on the number and type of the connected components of its meander. A slightly modified formula yields the index of a seaweed subalgebra of $\mathfrak{sl}(n)$ (see [3]). In the maximal parabolic case in type A, but using different methods, Elashvili [9] provided an explicit index formula which is presented in terms of a linear greatest common divisor of two arguments, each of which is a linear combination of the terms in the seaweed’s defining compositions. In [3, 5], Coll et al. developed a similar index formula in the next most complicated case – a total of four terms in the defining compositions – and conjectured that no single linear greatest common divisor formula could deliver the index of a general seaweed with more than four total terms in its defining compositions. In [13], Karnauhova and Liebscher proved this conjecture by establishing the following beautiful general theorem.

To set the notation, let $\underline{a}=(a_{1},\dots,a_{m})$ and $\underline{b}=(b_{1},\dots,b_{l})$ be two compositions of $n$, and let $M^{\rm A}_{n}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ denote the meander associated with the type-A seaweed $\mathfrak{p}^{\rm A}_{n}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$.

Theorem 1.1** (Karnauhova and Liebscher [13], 2015).**

If $m\geq 4$, then there do not exist homogeneous polynomials $f_{1},f_{2}\in\mathbb{Z}[x_{1},\dots,x_{m}]$ of arbitrary degree such that the number of connected components of $M^{\rm A}_{n}(\underline{a},n)$ is given by

[TABLE]

Extending this line of inquiry in [4], Coll et al. obtained similar combinatorial index formulas in the type-C case, $C_{n}=\mathfrak{sp}(2n)$. The requisite type-C meander development was undertaken by Coll, Hyatt, and Magnant in [4], where such meanders were called symplectic meanders. Somewhat later, these type-C meanders were developed independently by Panyushev and Yakimova (see [15]). However, the approach taken by Coll et al. is distinguished by an emphasis on closed-form linear index formulas and an analysis of a special set of vertices in a type-C meander which they called the tail of the meander. The analogue of the above theorem of Karnauhova and Liebscher in the type-C case is implicit in [4]. Moreover, it follows from Joseph ([12], Theorem 8.4) that the meandric index analysis in type-C carries over mutatis mutandis to the type-B case.

In this paper, we consider the type-D case, $D_{n}=\mathfrak{so}(2n)$. We follow the program outlined above and develop the following:

Type-D meanders. As with type C, our approach once again parallels the work of Panyushev and Yakimova in [16], but is distinguished, as before, by our goals and methods. We find, in particular, that the tail of the meander associated with a type-D seaweed has a more subtle structure. By exploiting the various configurations of the tail in this classical type, we develop a combinatorial formula for the index of a type-D seaweed based on the number and type of connected components in the seaweed’s associated meander. This formula is a bit less complicated than that found in [16] and can be used to classify index zero (Frobenius111Frobenius algebras are of special interest in deformation and quantum group theory stemming from their connection with the classical Yang-Baxter equation (see [10] and [11]). More specifically, an index-realizing functional is called regular, and a regular functional $F$ on a Frobenius Lie algebra $\mathfrak{g}$ is called a Frobenius functional; equivalently, $B_{F}(-,-)$ is non-degenerate. Suppose $B_{F}(-,-)$ is non-degenerate and let $[F]$ be the matrix of $B_{F}(-,-)$ relative to some basis $\{x_{1},\dots,x_{n}\}$ of $\mathfrak{g}$. In [1], Belavin and Drinfeld showed that

$\sum_{i,j}[F]^{-1}_{ij}x_{i}\wedge x_{j}$

is the infinitesimal of a Universal Deformation Formula (UDF) based on $\mathfrak{g}$. A UDF based on $\mathfrak{g}$ can be used to deform the universal enveloping algebra of $\mathfrak{g}$ and also the function space on any Lie group which contains $\mathfrak{g}$ in its Lie algebra of derivations.) type-D seaweeds up to a similarity transformation; 2. 2.

Obtain an exhaustive list of closed- form index formulas in all “reasonable cases”. Interestingly, seaweeds in type D are not necessarily seaweed “shaped” in their natural matrix representations. We discern why and and show that such algebras have the same index as a certain seaweed algebra of the same dimension that does have seaweed shape. 3. 3.

Establish the analogue in type D of the above theorem of Karnauhova and Liebscher (see Theorem 5.40).

The first four sections of the paper recount, and expand upon, the meander-based formulas in the first three classical families. We include these abridged results since we require them in their entirety to deal with the subtleties encountered in the type-D case.

The structure of the paper is as follows. In the single paragraph which comprises Section 2, we provide the formal definition of a seaweed algebra. Section 3 consists of brief summary of the results in type A, while Section 4 summarizes the type-C and type-B index results of Coll et al. (see [4].) Section 5 contains the main results of the paper, where the type-D case is analyzed.

2 Seaweeds

We assume that a seaweed $\mathfrak{g}$ is equipped with a triangular decomposition

[TABLE]

where $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, and $\mathfrak{u_{+}}$ and $\mathfrak{u_{-}}$ are the subalgebras consisting of the upper and lower triangular matrices, respectively. Let $\Pi$ be the set of $\mathfrak{g}$’s simple roots, and for $\alpha\in\Pi$, let $\mathfrak{g}_{\alpha}$ denote the root space corresponding to $\alpha$. A seaweed subalgebra $\mathfrak{p}\cap\mathfrak{p^{\prime}}$ is called standard if $\mathfrak{p}\supseteq\mathfrak{h}\oplus\mathfrak{u}_{+}$ and $\mathfrak{p^{\prime}}\supseteq\mathfrak{h}\oplus\mathfrak{u_{-}}$. In the case that $\mathfrak{p}\cap\mathfrak{p^{\prime}}$ is standard, let $\Psi=\{\alpha\in\Pi:\mathfrak{g}_{-\alpha}\notin\mathfrak{p}\}$ and $\Psi^{\prime}=\{\alpha\in\Pi:\mathfrak{g}_{\alpha}\notin\mathfrak{p^{\prime}}\}$, and denote the seaweed by $\mathfrak{p}(\Psi\hskip 2.84544pt|\hskip 2.84544pt\Psi^{\prime})$. Any seaweed is conjugate, over its algebraic group, to a standard one, so it suffices to work with standard seaweeds only. Note that an arbitrary seaweed may be conjugate to more than one standard seaweed (see [14], page 226).

3 Type A - $\mathfrak{sl}(n)$

3.1 Type-A seaweeds

Let $\mathfrak{sl}(n)$ be the algebra of $n\times n$ matrices with trace zero and consider the triangular decomposition of $\mathfrak{sl}(n)$ as above. Let $\Pi=\{\alpha_{1},\dots,\alpha_{n-1}\}$ be the set of simple roots of $\mathfrak{sl}(n)$ with the standard ordering, and let $\mathfrak{p}_{n}^{\rm A}(\Psi\hskip 2.84544pt|\hskip 2.84544pt\Psi^{\prime})$ denote a seaweed subalgebra of $\mathfrak{sl}(n)$, where $\Psi$ and $\Psi^{\prime}$ are subsets of $\Pi$.

Let $C_{n}$ denote the set of strings of positive integers whose sum is $n$. It will be convenient to index seaweeds of $\mathfrak{sl}(n)$ by pairs of elements of $C_{n}$. Let $\mathcal{P}(X)$ denote the power set of a set $X$. Let $\varphi_{\rm A}$ be the usual bijection from $C_{n}$ to a set of cardinality $n-1$. That is, given $\underline{a}=(a_{1},a_{2},\dots,a_{m})\in C_{n}$, define $\varphi_{\rm A}:C_{n}\rightarrow\mathcal{P}(\Pi)$ by

[TABLE]

Then define

[TABLE]

By construction, the sequence of numbers in $\underline{a}$ determines the heights of triangles below the main diagonal in $\mathfrak{p}_{n}^{\rm A}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ which may have nonzero entries, and the sequence of numbers in $\underline{b}$ determines the heights of triangles above the main diagonal. For example, the seaweed $\mathfrak{p}_{7}^{\rm A}((4,3)\hskip 2.84544pt|\hskip 2.84544pt(2,2,2,1))=\mathfrak{p}_{7}^{\rm A}(\{\alpha_{4}\}\hskip 2.84544pt|\hskip 2.84544pt\{\alpha_{2},\alpha_{4},\alpha_{6}\})$ has the following shape, where * indicates a possible nonzero entry. See the left-hand side of Figure 1.

Remark 3.1**.**

*The seaweed in Figure 1 has *seaweed shape: *Let $D_{\underline{a}}$ be the subalgebra of block-diagonal matrices whose blocks have sizes $a_{1}\times a_{1},\dots,a_{m}\times a_{m}$ and similarly for $D_{\underline{b}}$. A seaweed in type A has seaweed shape if it is the subalgebra of $\mathfrak{gl}(n)$ spanned by the intersection of $D_{\underline{a}}$ with the lower triangular matrices, the intersection of $D_{\underline{b}}$ with the upper triangular matrices, and all diagonal matrices. *

3.2 Type-A meanders

Given a seaweed $\mathfrak{p}_{n}^{\rm A}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ in $\mathfrak{sl}(n)$, Dergachev and A. Kirillov [6] showed how to associate a planar graph called a meander, denoted $M_{n}^{\rm A}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$. We label the vertices of $M_{n}^{\rm A}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ as $1,2,\dots,n$ from left to right, and place edges above them, called top edges, according to $\underline{a}$ as follows. Let $\underline{a}=(a_{1},a_{2},\dots,a_{m})$, and let $V_{i}$ be the $i^{\text{th}}$ block of vertices, that is the subset of vertices whose label is greater than $a_{1}+a_{2}+\dots+a_{i-1}$ and less than $a_{1}+a_{2}+\dots+a_{i}+1$. For each block $V_{i}$, place top edges connecting vertex $j$ to vertex $k$ if $j+k=2(a_{1}+a_{2}+\dots+a_{i-1})+a_{i}+1$. In the same way, place bottom edges according to $\underline{b}$. See the right-hand side of Figure 1.

Since each vertex is incident with at most one top edge, and at most one bottom edge, we define a top bijection $t$ on $\{1,...,n\}$ by $t(j)=k$ if there is a top edge from vertex $j$ to vertex $k$, and $t(j)=j$ if vertex $j$ is not incident with a top edge. Similarly, we define a bottom bijection $b$ on $\{1,...,n\}$. Given a meander $M_{n}^{\rm A}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$, let $\sigma_{\underline{a},\underline{b}}$ be its associated permutation defined by $\sigma_{\underline{a},\underline{b}}(j)=t(b(j))$. For example if $\underline{a}=(4,3)$ and $\underline{b}=(2,2,2,1)$, then the associated permutation written as a product of disjoint cycles is $\sigma_{\underline{a},\underline{b}}=(1,3)(2,4)(6,7,5)$.

The following result follows immediately from Theorem 5.1 of [6]. (Note that the formula in the theorem below differs by one from the theorem of Dergachev and Kirillov – since we are working in $\mathfrak{sl}(n)$ instead of $\mathfrak{gl}(n)$.)

Theorem 3.2**.**

The index of $\mathfrak{p}_{n}^{\rm A}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ with associated meander $M_{n}^{\rm A}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ is equal to $2C+P-1$, where $C$ is the number of cycles in $M_{n}^{\rm A}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ and $P$ is the number of paths in $M_{n}^{\rm A}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$.

An immediate consequence of Theorem 3.2 is that $\mathfrak{p}_{n}^{\rm A}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ is Frobenius if and only if $M_{n}^{\rm A}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ is a single path. For these seaweeds, the permutation $\sigma_{\underline{a},\underline{b}}$ is a single cycle and hence defines a permutation of $\{1,...,n\}$. The seaweed $\mathfrak{p}_{7}^{\rm A}((4,3)\hskip 2.84544pt|\hskip 2.84544pt(7))$ has associated permutation $\sigma_{\underline{a},\underline{b}}=(4,1,5,2,6,3,7)$.

3.3 Type-A index formulas

While Theorem 3.2 provides an elegant formalism for computing the index of a seaweed, significant computational complexity persists. What is needed is a mechanism for determining the index of a seaweed directly from its defining compositions. The first result of this kind is due to Elashvili, who used different notation to establish the following.

Theorem 3.3** (Elashvili [9], 1990).**

The seaweed $\mathfrak{p}_{n}^{\rm A}((a,b)\hskip 2.84544pt|\hskip 2.84544pt(n))$ has index $\rm{gcd}(a,b)-1.$

In [5], Coll et al. provide a recursive classification of meander graphs, showing that each meander is identified by a unique sequence of fundamental graph theoretic moves, each of which is uniquely determined by the structure of the meander at the time of move application. The sequence of (winding-down) moves is called the signature of the meander (see Appendix A, Lemma 6.1). Although discovered independently, the signature may be regarded as a graph theoretic rendering of Panyushev’s well-known reduction [14]. Using the signature, Coll et al. established the following extension of Elashvili’s theorem.

Theorem 3.4** (Coll et al. [5], 2015).**

The seaweeds $\mathfrak{p}_{n}^{\rm A}((a,b,c)\hskip 2.84544pt|\hskip 2.84544pt(n))$ and $\mathfrak{p}_{n}^{\rm A}((a,b)\hskip 2.84544pt|\hskip 2.84544pt(c,n-c))$ have index

[TABLE]

Remark 3.5**.**

The winding-down moves can be reversed to yield “winding-up” moves, which can be used to build any meander of any size and configuration (see Appendix A, Lemma 6.4).

One might conjecture the existence of similar “closed-form” index formulas for more general seaweeds, but, using signature moves and complexity arguments, Karnauhova and Liebscher have shown that there are severe restrictions.

Theorem 3.6** (Karnauhova and Liebscher [13], 2015).**

If $m\geq 4$, then there do not exist homogeneous polynomials $f_{1},f_{2}\in\mathbb{Z}[x_{1},\dots,x_{m}]$ of arbitrary degree such that the number of connected components of $M^{A}_{n}((a_{1},\dots,a_{m})\hskip 2.84544pt|\hskip 2.84544pt(n))$ is given by $\gcd(f_{1}(a_{1},\dots,a_{m}),f_{2}(a_{1},\dots,a_{m}))$.

4 Type C - $\mathfrak{sp}(2n)$ and Type B - $\mathfrak{so}(2n+1)$

4.1 Type-C seaweeds

In this subsection, we introduce type-C seaweeds and, following Coll et al. in [5], we develop type-C meanders (see also [15]). As in type A, the index of a type-C seaweed can be computed from simple graph theoretic properties of the type-C meander.

Let $\mathfrak{s}p(2n)$ be the algebras of matrices with the following block form

[TABLE]

where $A,B,$ and $C$ are $n\times n$ matrices, and $\widehat{A}$ is the transpose of $A$ with respect to the antidiagonal. Choose the same triangular decomposition as was done in the $\mathfrak{sl}(n)$ case, that is $\mathfrak{sp}(2n)=\mathfrak{u_{+}}\oplus\mathfrak{h}\oplus\mathfrak{u_{-}}$. Let $\Pi=\{\alpha_{1},\dots,\alpha_{n}\}$ denote its set of simple roots, where $\alpha_{n}$ is the exceptional root. Let $\mathfrak{p}_{n}^{\rm C}(\Psi\hskip 2.84544pt|\hskip 2.84544pt\Psi^{\prime})$ denote a seaweed subalgebra of $\mathfrak{sp}(2n)$, where $\Psi$ and $\Psi^{\prime}$ are subsets of $\Pi$.

Let $C_{\leq n}$ denote the set of strings of positive integers whose sum is less than or equal to $n$, and call each integer in the string a part. We will index seaweeds in $\mathfrak{sp}(2n)$ by pairs of elements from $C_{\leq n}$. Let $\mathcal{P}(X)$ denote the power set of a set $X$. Given $\underline{a}=(a_{1},a_{2},\dots,a_{m})\in C_{\leq n}$, define a bijection $\varphi_{\rm C}:C_{\leq n}\rightarrow\mathcal{P}(\Pi)$ by

[TABLE]

and define

[TABLE]

Example 4.1**.**

The seaweed $\mathfrak{p}_{3}^{\rm C}(\{\alpha_{3}\}\hskip 2.84544pt|\hskip 2.84544pt\{\alpha_{1}\})=\mathfrak{p}_{3}^{\rm C}((3)\hskip 2.84544pt|\hskip 2.84544pt(1))$ is the algebra of matrices in $\mathfrak{sp}(6)$ of the form in Figure 2 below, where * indicates a possible nonzero entry.

Remark 4.2**.**

Similar to the type-A case (see Remark 3.1), type-C seaweeds have seaweed shape: Let $D_{\underline{a}}$ be the subalgebra of block-diagonal matrices whose blocks have sizes $a_{1}\times a_{1},\dots,a_{m}\times a_{m},2(n-\sum a_{i})\times 2(n-\sum a_{i}),a_{m}\times a_{m},\dots,a_{1}\times a_{1}$ and similarly for $D_{\underline{b}}$. A type-C seaweed has seaweed shape if it is the subalgebra of $\mathfrak{gl}(n)$ spanned by the intersection of $D_{\underline{a}}$ with the lower triangular matrices, the intersection of $D_{\underline{b}}$ with the upper triangular matrices, and all diagonal matrices.

4.2 Type-C meanders

Given a seaweed $\mathfrak{p}_{n}^{\rm C}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$, associate a type-C meander, which we denote $M_{n}^{\rm C}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$. The construction is the same as type-A meanders. But for a type-C meander, we designate a special subset of vertices $T=T_{n}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ called the* tail* of the meander as follows: if $\underline{a}\in C_{\leq n}$, let $r=\sum a_{i}$, and define a subset of vertices $T_{n}(\underline{a})=\{v_{r+1},v_{r+2},\dots,v_{n}\}$. Then $T_{n}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ is the symmetric difference of $T_{n}(\underline{a})$ and $T_{n}(\underline{b})$, i.e.,

[TABLE]

Remark 4.3**.**

Note that if $\sum a_{i}\geq\sum b_{i}$, then $T=T_{n}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})=T_{n}(\underline{b})\setminus T_{n}(\underline{a})$. For convenience, we will assume $\sum a_{i}\geq\sum b_{i}$ for the remainder of this paper.

Example 4.4**.**

The type-C meander $M_{12}^{\rm C}((2,1,2,6)\hskip 2.84544pt|\hskip 2.84544pt(3,2,1,2))$ has tail $T=\{v_{9},v_{10},v_{11}\}$, indicated by yellow vertices in Figure 3.

The following theorem is the type-C analogue of the combinatorial formula for the index of type-A seaweeds given in Theorem 3.2.

Theorem 4.5** (Coll et al. [4], Theorem 4.5).**

Consider the seaweed $\mathfrak{p}_{n}^{\rm C}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$, and let $T=T_{n}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$. The index of $\mathfrak{p}_{n}^{\rm C}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ is equal to $2C+\widetilde{P}$ where $C$ is the number of cycles in $M_{n}^{\rm C}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ and $\widetilde{P}$ is the number of connected components containing either zero or two vertices from $T$ in $M_{n}^{\rm C}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$.

Example 4.6**.**

In Example 4.4, $C=1$ and $\widetilde{P}=3$, so ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{12}^{\rm C}((2,1,2,6)\hskip 2.84544pt|\hskip 2.84544pt(3,2,1,2))=5$.

The tail allows us to completely classify Frobenius type-C seaweeds up to similarity. The combinatorial formula in Theorem 4.5 is zero when $C$ and $\widetilde{P}$ are both zero, i.e., when all components of the meander are paths with one end in the tail. We record this in the following corollary.

Corollary 4.7**.**

A type-C seaweed is Frobenius if and only if its corresponding meander is a forest rooted in the tail.

Example 4.8**.**

The seaweed $\mathfrak{p}_{14}^{\rm C}\frac{7\hskip 2.84544pt|\hskip 2.84544pt7}{11}$ is Frobenius by Corollary 4.7. Below is its meander, with components highlighted.

The following corollary gives a necessary condition for a symplectic seaweed to have minimal index.

Corollary 4.9** (Coll et al. [4], Corollary 4.7).**

If ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm C}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})=0$, then $\sum a_{i}=n$, and $\sum b_{i}=n-r<n$, and there must be exactly $r$ odd integers among $\underline{a}$ and $\underline{b}$.

4.3 Type-C index formulas

Next, we consider type-C seaweeds where $\underline{a}$ and $\underline{b}$ have a small number of parts. Theorem 5.5 in [14] covers all cases when either $\underline{a}=\emptyset$ or $\underline{b}=\emptyset$. The next theorem considers the case when each of $\underline{a}$ and $\underline{b}$ has one part and contains a corrected typo from [4].

Corollary 4.10** (Coll et al. [4], Corollary 5.1).**

If $a=b$, then ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm C}((a)\hskip 2.84544pt|\hskip 2.84544pt(b))=n$. Otherwise,

[TABLE]

Now, we consider when $\underline{a}$ and $\underline{b}$ have a total of four parts. The following remark illustrates why we need not consider more complicated block configurations.

Remark 4.11**.**

Consider the type-C meander $M^{C}_{n}((a,b,c)\hskip 2.84544pt|\hskip 2.84544pt(d))$ where $a+b+c=n$ and $d<n$. The index computations for this meander are analogous to those for the meander $M^{A}_{2n-d}((n-d,a,b,c)\hskip 2.84544pt|\hskip 2.84544pt(2n-d))$ which by Theorem 3.6 has not just no linear gcd formula, but no polynomial gcd formula for its index.

We have the following three theorems for when $\underline{a}$ and $\underline{b}$ have a total of three parts.

Theorem 4.12** (Coll et al. [4], Theorm 5.2).**

Let $a+b=n$. If $c=n-1$ or $c=n-2$, then

[TABLE]

Theorem 4.13** (Coll et al. [4], Theorem 5.3).**

If $a+b=n$, then ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm C}((a,b)\hskip 2.84544pt|\hskip 2.84544pt(c))=0$ if and only if one of the following conditions hold:

1. (i)

$c=n-1$* and $\gcd(a+b,b+c)=1,$* 2. (ii)

$c=n-2$* and $\gcd(a+b,b+c)=1,$* 3. (iii)

$c=n-3$, the integers $a,b,$ and $c$ are all odd, and $\gcd(a+b,b+c)=2$.

Theorem 4.14** (Coll et al. [4], Theorem 5.6).**

The index of $\mathfrak{p}_{n}^{\rm C}((n)\hskip 2.84544pt|\hskip 2.84544pt(a,b))$ is equal to zero if and only if one of the following conditions hold:

1. (i)

$a+b=n-1$* and $\gcd(a+b,b+1)=1$,* 2. (ii)

$a+b=n-2$* and $\gcd(a+b,b+2)=1$,* 3. (iii)

$a+b=n-3$, the integers $n,a,$ and $b$ are all odd, and $\gcd(a+b,b+3)=2$.

Remark 4.15**.**

We could have alternatively used Theorem 4.13 or 4.14 (instead of Corollary 4.7) to conclude that the seaweed in Figure 4 is Frobenius.

5 Type D - $\mathfrak{so}(2n)$

Type-D seaweeds share some similarities with seaweeds in the types-B and C cases. However, in their standard representations, type-D seaweeds do not necessarily have seaweed shape. In Section 5.2, we discern what configuration of the defining roots causes this (see Theorem 5.1). When type-D seaweeds do have seaweed shape, their shape is exactly the same as in type C. In Section 5.3, to deal with seaweed-shaped seaweeds, we introduce (as with types B and C) the notion of a type-D meander and tail. In Section 5.4.1, we develop the type-D analogue of the combinatorial $2C+\widetilde{P}$ index formula (see Theorem 5.10). In Section 5.4.2, we leverage this combinatorics to yield linear gcd conditions for the index of certain type-D seaweeds based on the sizes of the parts that define them (see Theorem 5.27). In particular, we characterize Frobenius type-D seaweed-shaped seaweeds based on linear gcd conditions (and congruence properties) in the sizes of the parts that define the seaweed (see Theorems 5.34 and 5.38). We conclude the analysis of type-D seaweed-shaped seaweeds by establishing the type-D analogue of Theorem 3.6 of Karnauhova and Liebscher (see Theorem 5.40). In Section 5.5, we consider type-D seaweeds which do not have seaweed shape. We show that the index of a seaweed without seaweed shape can be computed by considering a seaweed which does have seaweed shape, and the index of the former and the latter differ by a constant (either 0 or 2). We use this to provide a classification of Frobenius type-D seaweeds which do not have seaweed shape (see Theorem 5.44).

5.1 Type-D seaweeds

Let $\mathfrak{s}o(2n)$ be the algebra of matrices with the following block form

[TABLE]

where $A,B,$ and $C$ are $n\times n$ matrices and $\widehat{A}$ is the transpose of $A$ with respect to the antidiagonal. Choose the same triangular decomposition as was done in the $\mathfrak{sl}(n)$ and $\mathfrak{sp}(2n)$ cases, that is $\mathfrak{so}(2n)=\mathfrak{u_{+}}\oplus\mathfrak{h}\oplus\mathfrak{u_{-}}$. Let $\Pi=\{\alpha_{1},\dots,\alpha_{n}\}$ denote its set of simple roots, where $\alpha_{n}$ is the exceptional root. Let $\mathfrak{p}_{n}^{\rm D}(\Psi\hskip 2.84544pt|\hskip 2.84544pt\Psi^{\prime})$ denote a seaweed subalgebra where $\Psi$ and $\Psi^{\prime}$ are subsets of $\Pi$. We find it convenient to visualize the seaweed by picturing the omitted roots. We call this the split Dynkin diagram for a seaweed. See Figure 5.

5.2 Type-D seaweed-shaped seaweeds

Curiously, type-D seaweeds do not necessarily have seaweed shape in their natural representation. Consequently, not all type-D seaweeds have the block triangular form which compositions can be obtained. We determine which type-D seaweeds do not have seaweed shape (see Theorem 5.1). We first examine type-D parabolics.

If $\mathfrak{p}$ is a parabolic subalgebra of $\mathfrak{so}(2n)$ defined by $\Psi\subseteq\Pi$ with $\alpha_{n-1}\not\in\Psi$ and $\alpha_{n}\in\Psi$, then $\mathfrak{p}$ does not have seaweed shape. However, if we remove $\alpha_{n}$ from and adjoin $\alpha_{n-1}$ to $\Psi$, then this yields an isomorphic parabolic which does have seaweed shape. See Figure 6.

All other type-D parabolics have seaweed shape. However, making this type of “switch” does not help for all type-D seaweeds, i.e., pairs of parabolics. In particular, the following theorem classifies the type-D seaweeds without seaweed shape.

Theorem 5.1**.**

Without loss of generality, $\mathfrak{p}_{n}^{\rm D}(\Psi\hskip 2.84544pt|\hskip 2.84544pt\Psi^{\prime})$ does not have seaweed shape if and only if $\alpha_{n-1}\in\Psi\setminus\Psi^{\prime}$ and $\alpha_{n}\in\Psi^{\prime}\setminus\Psi$.

Proof.

Let $\mathfrak{p}=\mathfrak{p}_{n}^{\rm D}(\Psi\hskip 2.84544pt|\hskip 2.84544pt\Psi^{\prime})$ with $\alpha_{n-1}\in\Psi\setminus\Psi^{\prime}$ and $\alpha_{n}\in\Psi^{\prime}\setminus\Psi$. Let $I=\max\{i\ |\ \alpha_{i}\in\Psi\}$. If $\Psi$ is empty, set $I=0$. The root spaces corresponding to $\alpha_{I+j}+\alpha_{I+j+1}+...+\alpha_{n-1}$ for $j=1,...,n-I-2$ are in $\mathfrak{p}_{1}$, and the root spaces corresponding to $\alpha_{I+j}+\alpha_{I+j+1}+...+\alpha_{n-2}+\alpha_{n}$ for $j=1,...,n-I-2$ are in $\mathfrak{p}_{2}$, but those corresponding to $\alpha_{I+j}+\alpha_{I+j+1}+...+\alpha_{n-1}$ for $j=1,...,n-I-2$ are not. So $\mathfrak{p}$ does not have seaweed shape.

For the converse, we make the following observation. Let $\psi\subseteq\{\alpha_{1},...\alpha_{n-2}\}$. Let $\Psi=\psi+\alpha_{n-1}$, and let $\Psi^{\prime}=\psi+\alpha_{n}$. Let $\mathfrak{p}_{1}$ be the parabolic subalgebra of $\mathfrak{so}(2n)$ determined by $\Pi\setminus\Psi$, and let $\mathfrak{p}_{2}$ be the parabolic subalgebra of $\mathfrak{so}(2n)$ determined by $\Pi\setminus\Psi^{\prime}$. Then $\mathfrak{p}_{1}\cong\mathfrak{p}_{2}$. The result follows. ∎

The following figure shows a type-D seaweed without seaweed shape.

Remark 5.2**.**

There is a nice visual representation for when a type-D seaweed does not have seaweed shape using a split Dynkin diagram: any of $\alpha_{1},...,\alpha_{n-2}$ can be included in either parabolic, indicated by the gray vertices, but the essential features occur are the bifurcation points.

While a seaweed without seaweed shape does not have block triangular form from which compositions can be obtained, type-D seaweeds with seaweed shape share this property with seaweeds of all other classical types. As in types B and C, there is a natural way to associate a partial composition of $n$ to each subset of simple roots defining a seaweed with seaweed shape. What is different from the types B and C cases, however, is that this association is not a bijection. Let $C_{\leq n}$ denote the set of strings of positive integers whose sum is less than or equal to $n$ and not equal to $n-1$, and (as before) call each integer in the string a part.

Remark 5.3**.**

In its natural representation, a seaweed with seaweed shape and $\alpha_{n-1}\not\in\Psi$ necessarily has $\alpha_{n}\not\in\Psi$; a seaweed $\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ with $\sum a_{i}=n-1$ and $\sum b_{i}\neq n-1$ is the same as the seaweed $\mathfrak{p}_{n}^{\rm D}(\underline{a},1\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$. We therefore exclude compositions of $n-1$ from our study.

Let $\mathcal{P}(X)$ denote the power set of a set $X$. Let $\Psi=\{\alpha_{s_{1}},\alpha_{s_{2}},\dots,\alpha_{s_{j}}\}\subseteq\mathcal{P}(\Pi)$, and assume $s_{1}<s_{2}<\cdots<s_{j}$, where if $\alpha_{n}\in\Psi$, then $\alpha_{n-1}\in\Psi$, define $\varphi_{\rm D}:\mathcal{P}(\Pi)\rightarrow C_{\leq n}$ by

[TABLE]

and define

[TABLE]

where $\underline{a}=\varphi_{\rm D}(\Psi)$ and $\underline{b}=\varphi_{\rm D}(\Psi^{\prime})$.

5.3 Type-D meanders

In this section, we introduce type-D meanders with the goal of creating type-D analogues of Theorem 3.2 and Theorem 4.5. A type-D meander is formed exactly as a type-C meander, and as in the type-C case, we will find it helpful to define a distinguished set of vertices called the tail of the meander. However, the type-D tail can take several “configurations”. The following critical definition sets the notation.

Definition 5.4**.**

Consider the seaweed $\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$. Assume $\sum a_{i}\geq\sum b_{j}$, and let $t=\sum a_{i}-\sum b_{j}$. We define the type-${\rm D}$ tail of $\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ to be

[TABLE]

We say that the tail, $T_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$, has configuration I, II, or III according to the three cases in (2). To ease notation, we will denote, for example, a seaweed $\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ with tail configuration III as $\mathfrak{p}_{n}^{\rm D}\left((\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b}),\textrm{III}\right)$, etc. When the compositions $\underline{a}$ and $\underline{b}$ are explicit, we will find it convenient to use the alternative fractional notation $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544pta_{m}}{b_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544ptb_{r}},\textrm{III}\right)$.

Example 5.5**.**

We illustrate the three cases in (2). The tail is indicated by yellow vertices.

5.4 Type-D formulas

In this section, we establish a combinatorial formula for the index of a type-D seaweed analogous to Theorem 3.2 and Theorem 4.5. We use this to classify Frobenius type-D seaweeds and extend these results to general index formulas.

5.4.1 Meander formula

The following three theorems give inductive formulas for the index. These will be used to prove the combinatorial formula in Theorem 5.10.

Theorem 5.6** (Panyushev and Yakimova [14], Theorem 5.2).**

Let $\underline{a}\neq\emptyset$ and $\underline{b}\neq\emptyset$. Consider the seaweed $\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$, where $\underline{a}=(a_{1},a_{2},\dots,a_{m})$ and $\underline{b}=(b_{1},b_{2},\dots,b_{r})$.

1. (i)

If $a_{1}=b_{1}$, then

[TABLE] 2. (ii)

If $a_{1}<b_{1}$, then

[TABLE]

Note that if $a_{1}>b_{1}$, we can use the fact that $\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})\cong\mathfrak{p}_{n}^{\rm D}(\underline{b}\hskip 2.84544pt|\hskip 2.84544pt\underline{a})$.

Theorem 5.7** (Dvorsky [8], Theorem 4.1).**

Let $\underline{a}=(a_{1},\dots,a_{m})$ with $\sum a_{i}=n$. For parabolic subalgebras of $\mathfrak{so}(2n)$,

1. (i)

if $n$ is even, then ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\emptyset)=\displaystyle\sum_{i=1}^{m}\left\lfloor\frac{a_{i}}{2}\right\rfloor$, 2. (ii)

if $n$ is odd, then

[TABLE]

Theorem 5.8** (Dvorsky [8], Theorem 4.3).**

Let $\underline{a}=(a_{1},\dots,a_{m})$ with $\sum a_{i}=a$. Let $k=n-a\geq 2$.

1. (i)

If $a$ is even, then ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\emptyset)=k+\displaystyle\sum_{i=1}^{m}\left\lfloor\frac{a_{i}}{2}\right\rfloor$. 2. (ii)

If $a$ is odd, then ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\emptyset)=k-1+\displaystyle\sum_{i=1}^{m}\left\lfloor\frac{a_{i}}{2}\right\rfloor$.

We have the following corollary of the above theorems. This is the type-D analogue of a type-C result used to prove the type-C combinatorial formula in Coll et al. [4].

Corollary 5.9**.**

Consider the seaweed $\mathfrak{p}_{n+k}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$, where $\underline{a}=(a_{1},a_{2},\dots,a_{m})$ and $\underline{b}=(b_{1},b_{2},\dots,b_{r})$. Suppose $\displaystyle n+k>n=\sum_{i=1}^{m}a_{i}\geq\sum_{i=1}^{r}b_{i}$. Let $\displaystyle t=\sum_{i=1}^{m}a_{i}-\sum_{i=1}^{r}b_{r}$.

1. (i)

If $t$ is even, then ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n+k}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})=k+{\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm C}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$. 2. (ii)

If $t$ is odd, then ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n+k}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})=k-1+{\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm C}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$.

Given a type-D meander $M_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$, we define top and bottom bijections $t$ and $b$ as before, and we associate to the meander the permutation $\sigma_{n,\underline{a},\underline{b}}$ defined by $\sigma_{n,\underline{a},\underline{b}}(j)=t(b(j))$. For example, if $\underline{a}=(4,3,2)$ and $\underline{b}=(2,2,2)$ are strings in $C_{\leq 9}$, then the associated permutation written in disjoint cycle form is $\sigma_{9,\underline{a},\underline{b}}=(1,3)(2,4)(5,7,6)(8,9)$. We can now establish the combinatorial index formula.

Theorem 5.10**.**

Consider the seaweed $\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$. Let $T=T_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$.

1. (i)

The index of $\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ is equal to $2C+\widetilde{P}$, where $C$ is the number of cycles in $M_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$, and $\tilde{P}$ is the number of paths containing either zero or two vertices from $T$ in $M_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$. 2. (ii)

The index of $\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ is equal to the number of cycles containing either zero or two integers from $T$ in the disjoint cycle decomposition of $\sigma_{n,\underline{a},\underline{b}}$ (here we view $T$ as a set of integers).

Proof.

A cycle in $M_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ cannot contain vertices from $T$, and breaks into two cycles in $\sigma_{n,\underline{a},\underline{b}}$. A path in $M_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ will be a cycle in $\sigma_{n,\underline{a},\underline{b}}$ containing the labels of all the vertices in the path. Thus, ($i$) and ($ii$) are equivalent, so it suffices to prove ($i$). By Corollary 5.9 and symmetry, it suffices to consider the case when $\sum a_{i}=n$ and $\sum b_{i}\leq n$. Now, induct on $n$. The base case is trivial. Given a meander $G$, let $f(G)$ denote the number of cycles plus the number of connected components containing either zero or two vertices from $T$ in $G$.

For the inductive step, first consider the case where $\underline{b}=\emptyset$. If $n$ is even, then all vertices belong to the tail $T$. If $n$ is odd, then all vertices except $v_{n}$ belong to $T$. There are no cycles in $M_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\emptyset)$. If $n$ is even, then there are no connected components containing zero vertices from $T$. If $n$ is odd and $a_{m}=1$, then there is one connected component containing zero vertices from $T$. If $n$ is odd and $a_{m}>1$, then there are no connected components containing zero vertices from $T$, and there is one connected component containing one vertex from $T$. Since each block of vertices $V_{i}$ is assigned $\left\lfloor a_{i}/2\right\rfloor$ top edges, it follows from Theorem 5.7 that

[TABLE]

To complete the inductive step, consider the case where $\underline{b}\neq\emptyset$. Suppose $a_{1}=b_{1}$. Let $H$ denote the subgraph of $M_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ induced by the vertices labeled 1 through $a_{1}$, and let $G$ denote the subgraph induced by the remaining vertices. Then $M_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})=H+G$ and $H$ contains no vertices from $T$. Clearly $f(H)=a_{1}$, and using the inductive hypothesis on $G$ we have

[TABLE]

By Theorem 5.6, the right-hand side of Equation (3) is equal to ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$.

Suppose $a_{1}\leq b_{1}/2$. By Theorem 6.1, the meander

[TABLE]

can be obtained from $M_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ by edge contractions that do not delete vertices from $T$. Thus, by induction, we have

[TABLE]

And by Theorem 5.6, the right-hand side of Equation (4) is equal to ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$.

Similarly, suppose $a_{1}>b_{1}/2$. By Theorem 6.1, the meander

[TABLE]

can be obtained from $M_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$ by edge contractions that do not delete vertices from $T$. Again, by induction, we have

[TABLE]

Again, by Theorem 5.6, the right-hand side of Equation (5) is equal to ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm D}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$. ∎

Example 5.11**.**

The seaweeds whose meanders are given by Figures 9, 10, and 11 have index one, two, and two, respectively.

With the established definition of the type-D tail, we can now concisely state a type-D analogue to the visual for type-C Frobenius seaweeds.

Theorem 5.12**.**

A type-D seaweed is Frobenius if and only if its corresponding meander graph is a forest rooted in the tail.

The following corollary reduces index computation for all type-D seaweeds with a tail of configuration I to previously-solved cases.

Theorem 5.13**.**

The index of $\mathfrak{p}_{n}^{\rm D}\left((\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b}),{\rm I}\right)$ equals the index of $\mathfrak{p}_{n}^{\rm C}(\underline{a}\hskip 2.84544pt|\hskip 2.84544pt\underline{b})$.

Remark 5.14**.**

As a corollary of Theorem 5.13, Theorem 4.12 holds when $c=n-2$, and case (ii) in each of Theorems 4.13 and 4.14 hold in type D.

We also have the following obstruction theorem to a type-D seaweed with tail of configuration ${\rm II}$ being Frobenius. In such seaweeds, the vertex $v_{n}$ is always a component separated from the tail.

Theorem 5.15**.**

A type-D seaweed with a tail of configuration ${\rm II}$ is never Frobenius.

5.4.2 Greatest common divisor formulas

In this subsection, we find explicit greatest common divisor formulas for the index in terms of elementary functions of the parts that determine the seaweed. We consider seaweeds where $\underline{a}$ and $\underline{b}$ have a small number of parts. Theorem 5.7 directly covers all cases when either $\underline{a}=\emptyset$ or $\underline{b}=\emptyset$. The next case we consider is when $\underline{a}$ and $\underline{b}$ each have one part. This case is easily handled by applying Theorem 5.6 of Panyushev and Theorem 5.7 of Dvorsky and should be considered a corollary of these results.

Theorem 5.16**.**

If $a=b$, then ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{n}^{\rm D}\frac{~{}a~{}}{~{}b~{}}=n$. Otherwise,

[TABLE]

The next case we consider is when $\underline{a}$ and $\underline{b}$ have a total of three parts. Having dispensed with configurations I and II in Theorems 5.13 and 5.15, respectively, we need only consider seaweeds of the form $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},\textrm{III}\right)$. In such seaweeds, the component containing $v_{n}$ can contribute to the index differently depending on how $b$ and $n-c$ are related, as the following figures illustrate. The tail is indicated by yellow vertices.

5.4.3 Seaweeds $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},\textrm{III}\right)$

The analysis of these seaweeds breaks into three cases, illustrated by the examples in Figures 12, 13, and 14, respectively.

**Case 1: ** $b=n-c$

If $b=n-c$, then $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},\textrm{III}\right)$ cannot be Frobenius since the components on the first $a$ vertices are always separated from the tail. Moreover, we have the following more general index formula.

Theorem 5.17**.**

If $b=n-c$, then

[TABLE]

**Case 2: ** $b<n-c$

The seaweed in Figure 13 is Frobenius by Theorem 5.12. Note that the subgraph on vertices $v_{1}$ through $v_{7}$ yields a Frobenius type-C meander. In general, when $v_{a}$ and $v_{a+1}$ are tail vertices, the meander can be separated into two parts: one on the first $a$ vertices, and the other on the last $b$ vertices, with no arc connecting the two subgraphs. We find that for such a seaweed to be Frobenius, it must have $b=2$ or $3$, a result which holds for general seaweeds of this form.

Theorem 5.18**.**

If $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544pta_{m}}{b_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544ptb_{r}},{\rm III}\right)$ is Frobenius and $a_{m}<n-\sum b_{i}$, then $a_{m}=2$ or $3$. Furthermore, $\mathfrak{p}_{n-a_{m}}^{\rm C}\dfrac{a_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544pta_{m-1}}{b_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544ptb_{r}}$ is Frobenius.

Proof.

If $a_{m}=1$, then $v_{n}$ is a path separated from the tail and contributes $1$ to the index. If $a_{m}\geq 4$, then the component containing $v_{n-1}$ is a path with two ends in the tail, which contributes one to the index. ∎

As a corollary of Theorems 5.16 and 5.18, we can classify which seaweeds in Case 2 are Frobenius.

Theorem 5.19**.**

The seaweed $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ with $b<n-c$ is Frobenius if and only if one of the following holds:

1. (i)

$b=2$* and $c=n-3$,* 2. (ii)

$b=3$, $c=n-5$, and $n$ is odd.

The following theorem gives a relation between type-C Frobenius seaweeds and type-D Frobenius seaweeds in Case 2. It serves as a partial converse to Theorem 5.18.

Theorem 5.20**.**

Let $\mathfrak{p}_{n}^{\rm C}\dfrac{a_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544pta_{m}}{b_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544ptb_{r}}$ be Frobenius with $\sum a_{i}>\sum b_{j}$.

1. (i)

If $\sum a_{i}-\sum b_{j}$ is odd, then $\mathfrak{p}_{2+n}^{\rm D}\left(\dfrac{a_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544pta_{m}\hskip 2.84544pt|\hskip 2.84544pt2}{b_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544ptb_{r}},{\rm III}\right)$ is Frobenius. 2. (ii)

If $\sum a_{i}-\sum b_{j}$ is even, then $\mathfrak{p}_{3+n}^{\rm D}\left(\dfrac{a_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544pta_{m}\hskip 2.84544pt|\hskip 2.84544pt3}{b_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544ptb_{r}},{\rm III}\right)$ is Frobenius.

Example 5.21**.**

The seaweed $\mathfrak{p}_{7}^{\rm C}\frac{4\hskip 2.84544pt|\hskip 2.84544pt3}{6}$ is Frobenius by Theorem 4.5. We can apply Theorem 5.20 to conclude $\mathfrak{p}_{9}^{\rm D}\frac{4\hskip 2.84544pt|\hskip 2.84544pt3\hskip 2.84544pt|\hskip 2.84544pt2}{6}$ in Figure 15 is Frobenius.

This example, together with Theorem 5.18, gives some insight into Frobenius seaweeds of the form $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a|b|k}{c},\textrm{III}\right)$ for specific $k$ with $k<n-c$.

Theorem 5.22**.**

The seaweed $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a|b|k}{c},{\rm III}\right)$ with $k=2$ or $3$ is Frobenius if and only if one of the following holds:

1. (i)

$k=2$, $c=n-3$, and $\gcd(a+b,b+c)=1$, 2. (ii)

$k=3$, $c=n-5$, and $\gcd(a+b,b+c)=1$, 3. (iii)

$k=2$, $c=n-5$, and $\gcd(a+b,b+c)=2$ with $a,b$, and $c$ all odd.

Proof.

This is immediate from Theorems 4.13 and 5.18. ∎

**Case 3: ** $b>n-c$

If $b>n-c$, then the tail, $T$, of a Frobenius seaweed $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},\textrm{III}\right)$ must have limited size.

Theorem 5.23**.**

If $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ is Frobenius, then $c=n-3$ or $c=n-5$. In particular, $|T|=2$ or $4$.

Proof.

Suppose, for a contradiction, that $c\geq n-7$. Then $\mathfrak{p}_{n-1}^{\rm C}\dfrac{a\hskip 2.84544pt|\hskip 2.84544pt1\hskip 2.84544pt|\hskip 2.84544ptb-2}{c}$ is Frobenius by Theorem 4.5. By Theorem 4.9, the seaweed $\mathfrak{p}_{n-1}^{\rm C}\dfrac{a\hskip 2.84544pt|\hskip 2.84544pt1\hskip 2.84544pt|\hskip 2.84544ptb-2}{c}$ must have exactly $n-1-c\geq 6$ odd integers among its parts – a contradiction. When $c=n-3$, the tail of $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ is given by $T=\{v_{n-2},v_{n-1}\}$. When $c=n-5$, the tail of $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ is given by $T=\{v_{n-4},v_{n-3},v_{n-2},v_{n-1}\}$. ∎

Remaining in Case 3, we now examine different tail sizes.

Case 3.1: $|T|=2$

We say the seaweed $\mathfrak{p}_{n}^{\rm D}\dfrac{a_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544pta_{m}}{b_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544ptb_{r}}$ has type-A homotopy type $H(k)$ if $\mathfrak{p}_{n}^{\rm A}\dfrac{a_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544pta_{m}\hskip 2.84544pt|\hskip 2.84544ptn-\sum a_{i}}{b_{1}\hskip 2.84544pt|\hskip 2.84544pt...\hskip 2.84544pt|\hskip 2.84544ptb_{r}\hskip 2.84544pt|\hskip 2.84544ptn-\sum b_{j}}$ has homotopy type $H(k)$. To find a type-D seaweed’s type-A homotopy type, we simply add an additional part (if necessary) to each partial composition and consider two full compositions of $n$.

Theorem 5.24**.**

If $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ is Frobenius, then it has type-A homotopy type $H(3)$ or $H(1)$.

Proof.

Such a seaweed consists of two paths: one containing $v_{n-2}$ and one containing $v_{n-1}$. Exactly one contains $v_{n}$. If $v_{n-2}$ and $v_{n}$ are on the same path, then $\mathfrak{p}_{n}^{\rm A}\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c\hskip 2.84544pt|\hskip 2.84544pt3}$ has homotopy type $H(3)$. If $v_{n-1}$ and $v_{n}$ are on the same path, then $\mathfrak{p}_{n}^{\rm A}\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c\hskip 2.84544pt|\hskip 2.84544pt3}$ has homotopy type $H(1)$. ∎

Example 5.25**.**

The following figures illustrate the two type-A homotopy types described in Theorem 5.24. In each figure, the meander without the dotted lower arc is a type-D meander; with the dotted lower arc included, it is the type-A meander from which the type-A homotopy type is discerned.

Letting $d=k$ in Theorem 5.2 of [2] gives the following useful corollary.

Theorem 5.26**.**

The seaweed $\mathfrak{p}=\mathfrak{p}_{n}^{\rm A}\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c\hskip 2.84544pt|\hskip 2.84544ptk}$ has homotopy type $H(k)$ if and only if

[TABLE]

Futhermore, $a$, $b$ and $c$ are all multiples of $k$. If $\hat{a}=\frac{a}{k}$, $\hat{b}=\frac{b}{k}$, and $\hat{c}=\frac{c}{k}$, then $M_{n}^{\rm A}\dfrac{\hat{a}\hskip 2.84544pt|\hskip 2.84544pt\hat{b}}{\hat{c}\hskip 2.84544pt|\hskip 2.84544pt1}$ consists of a single path.

We have the following theorem as a corollary.

Theorem 5.27**.**

If $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ has type-A homotopy type $H(3)$, then it is Frobenius if and only if $\gcd(a+b,b+c)=3$.

If $\mathfrak{p}_{n}^{\rm A}\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c\hskip 2.84544pt|\hskip 2.84544pt3}$ has homotopy type $H(1)$, then $\gcd(a+b,b+c)=1$. However, this is not enough to guarantee $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ is Frobenius as the following example illustrates.

Example 5.28**.**

The seaweed $\mathfrak{p}_{10}^{\rm D}\left(\dfrac{6\hskip 2.84544pt|\hskip 2.84544pt4}{7},\textrm{III}\right)$ has $\gcd(6+4,4+7)=1$ but is not Frobenius. However, $\mathfrak{p}_{10}^{\rm A}\dfrac{6\hskip 2.84544pt|\hskip 2.84544pt4}{7\hskip 2.84544pt|\hskip 2.84544pt3}$ does indeed have homotopy type $H(1)$. See Figure 18.

Comparing this example to the seaweed in Figure 17, we notice that for such $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},\textrm{III}\right)$ to be Frobenius, we need a condition that guarantees vertices $v_{n-2}$ and $v_{n-1}$ are on different components. To find this condition, we start with the following theorem.

Theorem 5.29**.**

Let $\underline{a}=(a_{1},a_{2},\dots a_{k})$ and $\underline{b}=(n)$ be compositions of $n$. Let $\sigma=\sigma_{\underline{a},\underline{b}}$. Consider the sequence $d_{\underline{a},\underline{b}}=(\sigma(1)-1,\sigma(2)-2,\dots,\sigma(n)-n)$. Then

[TABLE]

where $\Delta_{j}^{a_{j}}$ means that $\Delta_{j}$ appears $a_{j}$ times consecutively in the sequence, and $\Delta_{j}=2(a_{1}+a_{2}+\dots+a_{j-1})+a_{j}-n$ for $j=1,2,\dots,n$. Moreover, $\Delta_{i}\neq\Delta_{j}$ for $i\neq j$ and $k\neq 2$.

Proof.

Since $\underline{b}=(n)$, clearly $b(i)=n+1-i$ for $i=1,2,\dots,n$.

Let $j$ be any integer such that $1\leq j\leq k$. Let $i$ be any integer such that

[TABLE]

(Note that as $j$ ranges from 1 to $k$, we are in fact considering all integers $i$ from 1 to $n$.) We claim that $\sigma(i)=2(a_{1}+a_{2}+\dots+a_{j-1})+a_{j}-n+i$.

First we note that

[TABLE]

and

[TABLE]

We therefore have

[TABLE]

In other words, $b(i)$ is in the $j^{\text{th}}$ part of the top composition. It follows that $b(i)+t(b(i))$ is a constant equal to the first index plus the last index in the $j^{\text{th}}$ part of the top composition. Thus

[TABLE]

Solving for $t(b(i))$, we have

[TABLE]

as claimed. The theorem follows directly from the claim. ∎

Example 5.30**.**

The Frobenius seaweed $\mathfrak{p}_{9}^{\rm A}\frac{2\hskip 2.84544pt|\hskip 2.84544pt3\hskip 2.84544pt|\hskip 2.84544pt4}{9}$ has the following meander (see left-hand side of Figure 19) with $\Delta_{1}=2$, which appears two times, $\Delta_{2}=7$, which appears three times, and $\Delta_{3}=5$, which appears four times. The top-bottom map defines, in the obvious way, a permutation on the set $S=\{1,\dots,9\}$ to yield the permutation cycle $\sigma_{S}=(4~{}9~{}2~{}7~{}5~{}3~{}8~{}1~{}6)$. Now, define the mapping d, which gives the difference (mod 9) between consecutive elements of $\sigma_{S}$. We include a loop on a vertex when the top-map or the bottom-map is the identity on that vertex.

When there are only two top parts, we have the following easy corollary.

Corollary 5.31**.**

In Theorem 5.29, if $k=2$, then $\Delta_{1}=\Delta_{2}\equiv a_{1}-n~{}(\bmod~{}n)$. Here, we let $\Delta\equiv a_{1}-n~{}(\bmod~{}n)$.

Example 5.32**.**

The Frobenius seaweed $\mathfrak{p}_{8}^{\rm A}\frac{3\hskip 2.84544pt|\hskip 2.84544pt5}{8}$ has the following meander with $\Delta_{1}=3$ and $\Delta_{2}=3$ and permutation cycle $\sigma_{S}=(2~{}5~{}8~{}3~{}6~{}1~{}4~{}7)$. Note that $\Delta=3$ “generates” $\sigma_{S}$. In this case, the mapping d gives the difference (mod 8) between consecutive elements of $\sigma_{S}$.

Remark 5.33**.**

For Frobenius seaweeds $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ with $a+b=n$ and $c<n$, we have

[TABLE]

For the remainder of this paper, we will use $\Delta\equiv a-c~{}(\bmod~{}n)$ for such seaweeds.

We are now in a position to distinguish between the Frobenius and non-Frobenius cases described in, for example, Figures 17 and 18. We find that coupling the necessary greatest common divisor condition with a congrunece relation will classify certain families of Frobenius seaweeds.

Theorem 5.34**.**

If $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ has type-A homotopy type $H(1)$, then $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ is Frobenius precisely when the following two conditions are met:

1. (i)

$\gcd(a+b,b+c)=1$, and 2. (ii)

$0<\displaystyle\frac{\Delta^{\varphi(n)-1}}{n}-\left\lfloor\frac{\Delta^{\varphi(n)-1}}{n}\right\rfloor$* $<0.5$. Here, $\varphi(n)$ is the Euler $\varphi$ function.*

Proof.

The first condition must be satisfied by previous observations.

Let $\sigma$ be the permutation cycle for $\mathfrak{p}_{n}^{\rm A}\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c\hskip 2.84544pt|\hskip 2.84544pt3}$. Since $\Delta$ generates $\sigma$, there are distinct $k_{1},k_{2}\in(0,n)$ with

[TABLE]

If $k_{1}>k_{2}$, then the path in the meander for $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ containing $v_{n-1}$ also contains $v_{n}$. In particular, if $k_{1}>k_{2}$, then $v_{n-2}$ and $v_{n-1}$ are on different components in $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$, which, as noted earlier, is the second condition necessary for $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ to be Frobenius. These equations simplify to

[TABLE]

Multiplying equations (3) and (4) by $\Delta^{\varphi(n)-1}$, and applying Euler’s Totient theorem, yields the following system:

[TABLE]

If $k_{1}>k_{2}$, then $-\Delta^{\varphi(n)-1}~{}(\bmod~{}n)>\Delta^{\varphi(n)-1}~{}(\bmod~{}n)$, which is true when $0\leq\Delta^{\varphi(n)-1}~{}(\bmod~{}n)\leq\frac{n}{2}$. But $k_{i}\neq 0$ for $i=1,2$, so $\Delta^{\varphi(n)-1}~{}(\bmod~{}n)\neq 0$. Moreover, $k_{1}\neq k_{2}$, so $\Delta^{\varphi(n)-1}~{}(\bmod~{}n)\neq\frac{n}{2}$. The result follows. ∎

Example 5.35**.**

The Frobenius seaweed $\mathfrak{p}_{10}^{\rm D}\left(\dfrac{4\hskip 2.84544pt|\hskip 2.84544pt6}{7},\textrm{III}\right)$ has $\Delta=7$. With $\varphi(10)=4$, we compute $\displaystyle\frac{\Delta^{\varphi(n)-1}}{n}-\left\lfloor\frac{\Delta^{\varphi(n)-1}}{n}\right\rfloor=\frac{7^{3}}{10}-\left\lfloor\frac{7^{3}}{10}\right\rfloor=.3<.5$.

The seaweed $\mathfrak{p}_{10}^{\rm D}\left(\dfrac{6\hskip 2.84544pt|\hskip 2.84544pt4}{7},\textrm{III}\right)$ is not Frobenius. For this seaweed, $\Delta=9$. With $\varphi(10)=4$, we compute $\displaystyle\frac{\Delta^{\varphi(n)-1}}{n}-\left\lfloor\frac{\Delta^{\varphi(n)-1}}{n}\right\rfloor=\frac{9^{3}}{10}-\left\lfloor\frac{9^{3}}{10}\right\rfloor=.9>.5$

As a scholium of the proof of Theorem 5.34, notice if a seaweed satisfies all hypotheses of Theorem 5.34, but $.5<\displaystyle\frac{\Delta^{\varphi(n)-1}}{n}-\left\lfloor\frac{\Delta^{\varphi(n)-1}}{n}\right\rfloor$ $<1$, then it is not Frobenius.

Case 3.2: $|T|=4$

When $|T|=4$ and $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ is Frobenius, there are restrictions on the parts $a,b$, and $c$.

Lemma 5.36**.**

If $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ is Frobenius, then $a,b,$ and $c$ are odd. In particular, $n$ is even.

Proof.

Such a meander contains four paths, and hence eight ends of paths. Five of these are provided by the tail, so the other three must come from the parts. ∎

Theorem 5.37**.**

If $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ is Frobenius, then $v_{n-2}$ and $v_{n}$ are on the same component.

Proof.

If $v_{n}$ and $v_{n-1}$ are on the same component, then there are adjacent vertices connected by a single path, which contradicts that the blocks are of odd size.

If $v_{n}$ and $v_{n-3}$ are on the same component, then there are four consecutive vertices among $v_{1},...,v_{n-6}$ with an edge joining the first and the fourth while the second and third vertices are not connected. This cannot happen.

If $v_{n}$ and $v_{n-4}$ are on the same component, then applying moves from Lemma 6.1 to the meander $M_{n}^{\rm A}\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c\hskip 2.84544pt|\hskip 2.84544pt5}$ leaves the “meander”, which is not a valid homotopy type. See Figure 21.

∎

As a corollary of Theorem 5.37, if the seaweed $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ is Frobenius, it must have type-A homotopy type $H(1,1)$. Such seaweeds necessarily have $\gcd(a+b,b+c)=2$; however, this does not provide a sufficient characterization as Figures 22 and 23 illustrate.

To differentiate between these, we make the following observations about the type-A meanders from the previous examples:

Each meander consists of two paths, a blue path and an orange path; the blue path spans the even vertices, and the orange path spans the odd vertices. 2. 2.

In each meander, the orange path contains two components of the meander for $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$, each of which is a path with one end in the tail and contributes [math] to the index of $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$. 3. 3.

In the meander in Figure 22, the blue path contains two components of the meander for the seaweed $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$, each of which is a path with one end in the tail and contributes zero to the index of $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$. However, the same does not hold for the meander in Figure 23. Here, while the blue path does contain two components of the meander for $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$, one is a path with zero ends in the tail, and the other is a path with two ends in the tail; each of these components contributes one to the index of $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$. 4. 4.

The top-bottom map gives two permutations associated to each meander: the first, $\sigma_{1}$ (the blue path), spanning the even vertices and a second, $\sigma_{2}$ (the orange path), spanning the odd vertices. Each permutation is generated by $\Delta\equiv a-c~{}(\bmod~{}n)$ by Theorem 5.29.

Combining observations $3$ and $4$ above, what will differentiate the Frobenius case from the non-Frobenius case must be captured by the blue path. We will add to the previously-mentioned greatest common divisor condition an argument similar to the proof of Theorem 5.34 to obtain a sufficient condition.

Theorem 5.38**.**

The seaweed $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ is Frobenius precisely when the following two conditions are met:

1. (i)

$\gcd(a+b,b+c)=2$, and 2. (ii)

$0<\displaystyle\frac{(\frac{\Delta}{2})^{\varphi(\frac{n}{2})-1}}{n}-\left\lfloor\frac{(\frac{\Delta}{2})^{\varphi(\frac{n}{2})-1}}{n}\right\rfloor$* $<.5$. Here, $\varphi(n)$ is the Euler $\varphi$ function.*

Proof.

The first condition must be satisfied by previous observations.

Let $\sigma_{1}$ be the permutation cycle spanning the even vertices in $\mathfrak{p}_{n}^{\rm A}\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c\hskip 2.84544pt|\hskip 2.84544pt5}$. Let $\sigma$ be the permutation of $\{1,...,\frac{n}{2}\}$ obtained by dividing each entry of $\sigma_{1}$ by $2$. Since $\Delta$ generates $\sigma_{1}$, $\frac{\Delta}{2}$ generates $\sigma$. Thus there are distinct $k_{1},k_{2}\in(0,\frac{n}{2})$ with

[TABLE]

If $k_{1}>k_{2}$, then the path in the meander for $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$ containing $v_{n-2}$ also contains $v_{n}$. In particular, if $k_{1}>k_{2}$, then $\mathfrak{p}_{n}^{\rm D}\left(\dfrac{a\hskip 2.84544pt|\hskip 2.84544ptb}{c},{\rm III}\right)$, is Frobenius. These equations simplify to

[TABLE]

If $k_{1}>k_{2}$ and $0\leq\frac{\Delta}{2}^{\varphi(\frac{n}{2})-1}~{}(\bmod~{}\frac{n}{2})\leq\frac{n}{4}$, then $-\frac{\Delta}{2}^{\varphi(\frac{n}{2})-1}~{}(\bmod~{}\frac{n}{2})>\frac{\Delta}{2}^{\varphi(\frac{n}{2})-1}~{}(\bmod~{}\frac{n}{2})$. But $k_{i}\neq 0$ for $i=1,2$, so $\Delta^{\varphi(\frac{n}{2})-1}~{}(\bmod~{}\frac{n}{2})\neq 0$. The result follows. ∎

Example 5.39**.**

The Frobenius seaweed $\mathfrak{p}_{14}^{\rm D}\left(\dfrac{5\hskip 2.84544pt|\hskip 2.84544pt9}{9},\textrm{III}\right)$ has $\frac{\Delta}{2}=5$. With $\varphi(7)=6$, we compute $\displaystyle\frac{(\frac{\Delta}{2})^{\varphi(\frac{n}{2})-1}}{n}-\left\lfloor\frac{(\frac{\Delta}{2})^{\varphi(\frac{n}{2})-1}}{n}\right\rfloor=\frac{5^{5}}{14}-\left\lfloor\frac{5^{5}}{14}\right\rfloor\approx.21<.5$.

The seaweed $\mathfrak{p}_{22}^{\rm D}\left(\dfrac{9\hskip 2.84544pt|\hskip 2.84544pt13}{17},\textrm{III}\right)$ is not Frobenius. For this seaweed, $\frac{\Delta}{2}=7$. With $\varphi(11)=10$, we compute $\displaystyle\frac{(\frac{\Delta}{2})^{\varphi(\frac{n}{2})-1}}{n}-\left\lfloor\frac{(\frac{\Delta}{2})^{\varphi(\frac{n}{2})-1}}{n}\right\rfloor=\frac{7^{9}}{22}-\left\lfloor\frac{7^{9}}{22}\right\rfloor\approx.86>.5$

As a scholium of the proof of Theorem 5.38, notice that if a seaweed satisfies every condition in Theorem 5.38, but $.5<\displaystyle\frac{(\frac{\Delta}{2})^{\varphi(n)-1}}{n}-\left\lfloor\frac{(\frac{\Delta}{2})^{\varphi(n)-1}}{n}\right\rfloor$ $<1$, then it is not Frobenius.

We now consider seaweeds of the form $\mathfrak{p}_{n}^{\rm D}\frac{a\hskip 2.84544pt|\hskip 2.84544ptb\hskip 2.84544pt|\hskip 2.84544ptc}{d}$ with $a+b+c=n$. If $d=n$, we obtain an index formula as a corollary to type-A results. But for $d<n$, we leverage Theorem 3.6 by Karnauhova and Liebscher to show that such seaweeds not only have no linear gcd formula for their index, but also no polynomial gcd formula for their index, regardless of tail configuration.

Theorem 5.40**.**

Consider the seaweed $\mathfrak{p}=\mathfrak{p}_{n}^{\rm D}\frac{a\hskip 2.84544pt|\hskip 2.84544ptb\hskip 2.84544pt|\hskip 2.84544ptc}{d}$ with $a+b+c=n$.

1. (i)

If $d=n$, then ${\rm ind\hskip 2.84544pt}\mathfrak{p}=\gcd(a+b,b+c)$. 2. (ii)

If $d<n$, then there do not exist homogeneous polynomials $f_{1},f_{2}\in\mathbb{Z}[x_{1},x_{2},x_{3}]$ of arbitrary degree such that ${\rm ind\hskip 2.84544pt}\mathfrak{p}$ is given by $\gcd(f_{1}(a,b,c),f_{2}(a,b,c))$.

Proof.

Let $M$ be the meander for $\mathfrak{p}$. If $d=n$, then $M$ contains zero tail vertices, so by Theorem 5.10, ${\rm ind\hskip 2.84544pt}\mathfrak{p}=2C+P$, where $P$ is the number of paths in $M$. Consider the seaweed $\mathfrak{p}_{1}=\mathfrak{p}_{n}^{\rm A}\frac{a\hskip 2.84544pt|\hskip 2.84544ptb\hskip 2.84544pt|\hskip 2.84544ptc}{d}$. By Theorem 3.4, ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{1}=\gcd(a+b,b+c)-1$. By Theorem 3.2, ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{1}=2C+P-1$. Hence $\gcd(a+b,b+c)=2C+P$. But the meander for $\mathfrak{p}_{1}$ is isomorphic to $M$, so ${\rm ind\hskip 2.84544pt}\mathfrak{p}=\gcd(a+b,b+c)$.

If $d<n$, then the index computations for $M$ have the same complexity as the index computations for $\mathfrak{p}^{A}_{2n-d}\frac{n-d\hskip 2.84544pt|\hskip 2.84544pta\hskip 2.84544pt|\hskip 2.84544ptb\hskip 2.84544pt|\hskip 2.84544ptc}{2n-d}$, which by Theorem 3.6 has no polynomial gcd formula for its index. ∎

Example 5.41**.**

The black seaweed in Figure 24 is a four-part type-D seaweed which can be extended to an arbitrary five-part type-A seaweed whose connected components cannot be counted by a polynomial gcd formula. Intuitively, a type-D seaweed with four total parts corresponds to a five-part type-A seaweed using a construction similar to that in Figure 24.

5.5 Type-D seaweeds without seaweed shape

Finally, we analyze type-D seaweeds without seaweed shape. Recall that the classification from Theorem 5.1 and Figure 8 provide a useful visual for such seaweeds. To compute the index of seaweeds without seaweed shape, we will make a specific switch in which simple roots define the seaweed, yielding a new seaweed which does have seaweed shape. The index of the original seaweed is either the same as the index of the new seaweed or the index of the new seaweed minus two. As a corollary of Theorem 4.1 in [16], we have the following theorem.

Theorem 5.42**.**

Let $\mathfrak{p}_{1}=\mathfrak{p}_{n}^{\rm D}(\Psi\hskip 2.84544pt|\hskip 2.84544pt\Psi^{\prime})$ be a seaweed without seaweed shape with $\alpha_{n-1}\in\Psi\setminus\Psi^{\prime}$. Let $\Psi^{\prime\prime}=\Psi^{\prime}-\alpha_{n}+\alpha_{n-1}$. Then $\mathfrak{p}_{2}=\mathfrak{p}_{n}^{\rm D}(\Psi\hskip 2.84544pt|\hskip 2.84544pt\Psi^{\prime\prime})=\mathfrak{p}_{n}^{\rm D}((a_{1},...,a_{m})\hskip 2.84544pt|\hskip 2.84544pt(b_{1},...,b_{l}))$ has seaweed shape. Let $M$ be the meander associated to $\mathfrak{p}_{2}$, and consider vertices $v_{n-a_{m}+1}$ and $v_{n}$ in $M$. Then

1. (i)

${\rm ind\hskip 2.84544pt}\mathfrak{p}_{1}={\rm ind\hskip 2.84544pt}\mathfrak{p}_{2}$* if $v_{n-a_{m}+1}$ and $v_{n}$ are on a path, and* 2. (ii)

${\rm ind\hskip 2.84544pt}\mathfrak{p}_{1}={\rm ind\hskip 2.84544pt}\mathfrak{p}_{2}-2$* if $v_{n-a_{m}+1}$ and $v_{n}$ are on a cycle.*

Proof.

This follows since $M$ has no tail vertices. ∎

Example 5.43**.**

We illustrate the two cases of Theorem 5.42 in Figures 25 and 26, respectively.

As a corollary of Theorem 5.42, we can classify Frobenius type-D seaweeds without seaweed shape.

Theorem 5.44**.**

There is a bijection between Frobenius type-D seaweeds without seaweed shape and type-A seaweeds with homotopy type $H(2)$.

Proof.

Following the notation of Theorem 5.42, since ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{2}\geq 0$, the only way ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{1}$ can equal zero is if distinguished vertices $v_{n-a_{k}+1}$ and $v_{n}$ are on a cycle and ${\rm ind\hskip 2.84544pt}\mathfrak{p}_{2}=2$. The only such $\mathfrak{p}_{2}$ have a meander graph consisting of exactly one cycle. Moreover, the parts defining $\mathfrak{p}_{2}$ also define a type-A seaweed, and consequently, a type-A meander identical to $M$. Such a type-A seaweed has homotopy type $H(2)$. ∎

6 Appendix A - The signature and homotopy type of a meander

The following lemma establishes that, using a deterministic sequence of graph-theoretic moves, each meander can be contracted or “wound down” to the empty meander, a meander with no vertices. The sequence of moves applied to a meander is called the signature of the meander, and the meander’s homotopy type can then be read off of the signature.

Lemma 6.1** (Coll et al. [2], Lemma 4.1).**

Given the meander $M=M_{n}^{\rm A}\frac{a_{1}|a_{2}|...|a_{m}}{b_{1}|b_{2}|...|b_{t}}$, a new meander $M^{\prime}$ can be created by one of the following moves:

1. (i)

Flip $(F)$:* If $a_{1}<b_{1}$, then $M^{\prime}=M_{n}^{\rm A}\frac{b_{1}|b_{2}|...|b_{t}}{a_{1}|a_{2}|...|a_{m}}$,* 2. (ii)

Component Elimination $(C(c))$:* If $a_{1}=b_{1}=c$, then $M^{\prime}=M_{n}^{\rm A}\frac{a_{2}|a_{3}|...|a_{m}}{b_{2}|b_{3}|...|b_{t}},$* 3. (iii)

Rotation Contraction $(R)$:* If $b_{1}<a_{1}<2b_{1}$, then $M^{\prime}=M_{n}^{\rm A}\frac{b_{1}|a_{2}|a_{3}|...|a_{m}}{(2b_{1}-a_{1})|b_{2}|...|b_{t}}$,* 4. (iv)

Block Elimination $(B)$:* If $a_{1}=2b_{1}$, then $M^{\prime}=M_{n}^{\rm A}\frac{b_{1}|a_{2}|..|a_{m}}{b_{2}|b_{3}|...|b_{t}}$,* 5. (v)

Pure Contraction $(P)$:* If $a_{1}>2b_{1}$, then $M^{\prime}=M_{n}^{\rm A}\frac{(a_{1}-2b_{1})|b_{1}|a_{2}|a_{3}|...|a_{m}}{b_{2}|b_{3}|...|b_{t}}$.*

This winding down process is illustrated by the following example.

Example 6.2**.**

We find the signature of the meander $M_{10}^{\rm A}\frac{3\hskip 2.84544pt|\hskip 2.84544pt7}{2\hskip 2.84544pt|\hskip 2.84544pt5\hskip 2.84544pt|\hskip 2.84544pt3}$ in Figure 27.

The component elimination moves in Lemma 6.1 give the homotopy type of the meander. A meander has homotopy type $H(a_{1},a_{2},\dots,a_{m})$ if its signature contains $C(a_{i})$ exactly once for all integers $i\in[1,m]$ in addition to no other component elimination moves. Further, we say if a meander has homotopy type $H(a_{1},a_{2},\dots,a_{m})$, then it is homotopically equivalent to the meander $M_{\sum a_{i}}^{\rm A}\dfrac{a_{1}|a_{2}|\dots|a_{m}}{a_{1}|a_{2}|\dots|a_{m}}$. We define the homotopy type of a seaweed to be the homotopy type of its corresponding meander.

Example 6.3**.**

The seaweed $\mathfrak{p}_{10}^{\rm A}\frac{3\hskip 2.84544pt|\hskip 2.84544pt7}{2\hskip 2.84544pt|\hskip 2.84544pt5\hskip 2.84544pt|\hskip 2.84544pt3}$ has homotopy type $H(1,3)$.

Note that each of the moves in Lemma 6.1 can be reversed to yield a “winding-up” move. These moves, which we record in the following lemma, can be used to build any meander of any size and block configuration.

Lemma 6.4** (Coll et al. [2], Lemma 4.2).**

Every meander is the result of a sequence of the following moves applied to the empty meander. Given the meander $M=M_{n}^{\rm A}\dfrac{a_{1}|a_{2}|\dots|a_{m}}{b_{1}|b_{2}|\dots|b_{t}}$, create a meander $M^{\prime}$ by one of the following moves:

1. (i)

Flip $(\widetilde{F})$:* $M^{\prime}=\displaystyle\frac{b_{1}|b_{2}|...|b_{t}}{a_{1}|a_{2}|...|a_{m}}$,* 2. (ii)

Component Creation $(\widetilde{C}(c))$:* $M^{\prime}=\displaystyle\frac{c|a_{1}|a_{2}|...|a_{m}}{c|b_{1}|b_{2}|...|b_{t}},$* 3. (iii)

Rotation Expansion $(\widetilde{R})$:* if $a_{1}>b_{1}$, then $M^{\prime}=\displaystyle\frac{(2a_{1}-b_{1})|a_{2}|\dots|a_{m}}{a_{1}|b_{2}|\dots|b_{t}}$,* 4. (iv)

Block Creation $(\widetilde{B})$:* $M^{\prime}=\displaystyle\frac{2a_{1}|a_{2}|\dots|a_{m}}{a_{1}|b_{1}|b_{2}|\dots|b_{t}},$* 5. (v)

Pure Expansion $(\widetilde{P})$:* $M^{\prime}=\displaystyle\frac{a_{1}+2a_{2}|a_{3}|a_{4}|\dots|a_{m}}{a_{2}|b_{1}|b_{2}|\dots|b_{t}}.$*

Remark 6.5**.**

All moves in Lemmas 6.1 and 6.4 preserve homotopy type except for the component elimination and the component creation moves.