Smooth Schubert varieties in the affine flag variety of type $\tilde{A}$

TL;DR

This paper proves that smooth Schubert varieties in affine type A are iterated Grassmannian bundles, confirming a conjecture about pattern avoidance characterizing smoothness and providing a generating function for their count.

Contribution

It extends finite type A results to affine type A, establishing the fiber bundle structure and confirming the pattern avoidance criterion for smoothness.

Findings

Smooth affine Schubert varieties are iterated Grassmannian bundles.

A conjecture linking smoothness to pattern avoidance is confirmed.

A generating function for counting smooth affine Schubert varieties is derived.

Abstract

We show that every smooth Schubert variety of affine type $\tilde{A}$ is an iterated fibre bundle of Grassmannians, extending an analogous result by Ryan and Wolper for Schubert varieties of finite type $A$. As a consequence, we finish a conjecture of Billey-Crites that a Schubert variety in affine type $\tilde{A}$ is smooth if and only if the corresponding affine permutation avoids the patterns $4231$ and $3412$. Using this iterated fibre bundle structure, we compute the generating function for the number of smooth Schubert varieties of affine type $\tilde{A}$.

Tables

Table 1. Table 1. Number of smooth Schubert varieties of type A ~ n subscript ~ 𝐴 𝑛 \tilde{A}_{n} .

$n=2$	$n=3$	$n=4$	$n=5$	$n=6$	$n=7$	$n=8$	$n=9$
5	31	173	891	4373	20833	97333	448663

Equations

$$A(t)=\frac{P(t)-Q(t)\sqrt{1-4t}}{(1-t)(1-4t)\left(1-6t+8t^2-4t^3\right)}$$

$$P(t)=(1-4t)\left(2-11t+18t^2-16t^3+10t^4-4t^5\right)$$

$$Q(t)=(1-t)(2-t)\left(1-6t+6t^2\right).$$

$$\alpha :=\frac{1}{6}\left(4-\sqrt[3]{17+3\sqrt{33}}+\sqrt[3]{-17+3\sqrt{33}}\right)\approx 0.228155$$

$$P_w^J(q)=\sum _{x\le w\text{ and }x\in W^J}{q}^{\ell (x)},$$

$$S(v^{\prime })\cap K^{\prime }\subseteq D_L(u^{\prime })\subseteq D_L(u^{\prime }v),$$

$$X^J(w)=X_m\to X_{m-1}\to \cdots \to X_1\to X_0=\text{pt},$$

$$x(i,k)=s_{i+k-1}{s}_{i+k-2}\cdots s_i\text{ and }y(i,k)=s_{i-k+1}{s}_{i-k+2}\cdots s_i,$$

$$X^J(w)=X_m\to X_{m-1}\to \cdots \to X_1\to X_0=\text{point},$$

$${\mathcal{D}}_s:=\{B\in \mathcal{D}|s\in B\}.$$

$$\overrightarrow{[s_i,s_j]}:=\{\begin{array}{ll}\{s_i,\dots ,s_j\}& \text{if i≤j}\\ \{s_i,\dots ,s_{n-1},s_0,\dots ,s_j\}& \text{if i>j}\end{array}$$

$$\mathcal{D}=\{\overrightarrow{[s_0,s_3]}\prec \overrightarrow{[s_7,s_1]}\prec \overrightarrow{[s_5,s_7]}\prec \overrightarrow{[s_3,s_6]}\}$$

$$s_0$$

$$s_1$$

$$\mathcal{D}=\{B_1\prec B_2\prec \cdots \prec B_m\}$$

$$A_M(t):=\sum _{n=1}^{\infty }{m}_n{t}^n$$

$$r(B_i):=\#\{s\in B_i\setminus B_{i-1}\}\text{and}u(B_i):=\#\{s\in B_i\setminus B_{i+1}\}$$

$$\sum _{i=1}^{m}r(B_i)=\sum _{i=1}^{m}u(B_i)=n$$

$$\sum _{k=1}^{i}r(B_k)\ge \sum _{k=1}^{i}u(B_k)$$

$${\bar{B}}_i:=\left\{s_j\in S|\sum _{k=1}^{i-1}{u}_k<j\le \sum _{k=1}^i{r}_k\right\}$$

$$\mathcal{B}=\{B\cap S_n|B\in \mathcal{D}\}$$

$$A_B(t)=\sum _{n=1}^{\infty }{b}_n{t}^n$$

$$t+t{A}_B(t)=A_M(t)-t{A}_M(t).$$

$$J_j=\{\begin{array}{ll}\overrightarrow{[s_{c_j},s_{c_{j+1}-1}]}& 1\le j<m\text{vspace}.1in\\ \overrightarrow{[s_{c_m},s_{c_1-1}]}& j=m\end{array},$$

$${\mathcal{B}}_j=\{B\cap J_j:B\in \mathcal{D}\text{ and }B\cap J_j\ne \varnothing \}$$

$$\mathcal{D}=\{\overrightarrow{[s_1,s_3]}\prec \overrightarrow{[s_3,s_4]}\prec \overrightarrow{[s_5,s_6]}\succ \overrightarrow{[s_6,s_7]}\succ \overrightarrow{[s_7,s_8]}\prec \overrightarrow{[s_9,s_1]}\}$$

$$\bar{A}(t)=\sum _{n=1}^{\infty }{\bar{a}}_n{t}^n$$

$$A_F(t):=\sum _{n=0}^{\infty }{f}_n{t}^n.$$

$$b_j=\{\begin{array}{ll}{a}_{j+1}-1& j<m\\ n& j=m\end{array}.$$

$$A(t)=\sum _{n=1}^{\infty }{a}_n{t}^n.$$

Full text

1. Introduction

Let $X$ be a Kac-Moody flag variety, and let $W$ be the associated Weyl group. Although $X$ can be infinite-dimensional, it is stratified by finite-dimensional Schubert varieties $X(w)$, where $w\in W$. It is natural to ask when $X(w)$ is smooth or rationally smooth, and this question is well-studied [BL00]. For the finite-type flag variety of type $A_{n}$, the Weyl group is the permutation group $\mathfrak{S}_{n}$,111Note that we use $\mathfrak{S}$ to refer to groups, and $S$ to refer to sets of simple reflections. and the Lakshmibai-Sandhya theorem states that $X(w)$ is smooth if and only if $w$ avoids the permutation patterns $3412$ and $4231$ [LS90]. From another angle, the Ryan-Wolper theorem states that $X(w)$ is smooth if and only if $X(w)$ is an iterated fibre bundle of Grassmannians of type $A$ [Rya87, Wol89]. Haiman used the Ryan-Wolper theorem to enumerate smooth Schubert varieties [Hai, Bón98]. The Lakshmibai-Sandhya theorem, the Ryan-Wolper theorem, and the enumeration of smooth and rationally smooth Schubert varieties has been extended to all finite types (see [Bil98, BP05], [RS16], and [RS15] respectively). The latter enumeration uses a data structure called staircase diagrams, which keeps track of iterated fibre bundle structures.

There are also characterizations of smoothness and rational smoothness that apply to all Kac-Moody types [Car94, Kum96]. For instance, a theorem of Carrell and Peterson states that $X(w)$ is rationally smooth if and only if the Poincare polynomial $P_{w}(q)$ of $X(w)$ is palindromic, meaning that the coefficients read the same from top-degree to bottom-degree and vice-versa [Car94]. However, much less is known about the structure of (rationally) smooth Schubert varieties in general Kac-Moody type. The one exception is affine type $\tilde{A}$, where Billey and Crites have characterized the elements $w$ for which $X(w)$ is rationally smooth [BC12]. In this case, the Weyl group $W$ is the affine permutation group $\tilde{\mathfrak{S}}_{n}$. As part of their characterization, they prove that if $X(w)$ is smooth, then $w$ must avoid the affine permutation patterns $3412$ and $4231$. They conjecture the converse, that $X(w)$ is smooth if $w$ avoids these two patterns.

The purpose of this paper is to extend what we know about finite-type Schubert varieties to affine type $\tilde{A}$. For smooth Schubert varieties, we show:

Theorem 1.1**.**

Let $X(w)$ be a Schubert variety in the full flag variety of type $\tilde{A}$. Then the following are equivalent:

1. (a)

$X(w)$* is smooth.* 2. (b)

$w$* avoids the affine permutation patterns $3412$ and $4231$.* 3. (c)

$X(w)$* is an iterated fibre bundle of Grassmannians of finite type $A$.*

In particular, this finishes the proof of Billey and Crites’ conjecture. We note that the proof relies heavily on ideas from both [BC12] and [RS16]. One corollary (explained in Section 3) is that there is a bijection between smooth Schubert varieties in the full flag variety of type $\tilde{A}_{n}$, and spherical staircase diagrams over the Dynkin diagram of type $\tilde{A}_{n}$. This allows us to enumerate smooth Schubert varieties in affine type $\tilde{A}_{n}$:

Theorem 1.2**.**

Let $A(t)=\sum a_{n}\,t^{n}$, where $a_{n}$ is the number of smooth Schubert varieties in the full flag variety of type $\tilde{A}_{n}$. Then

[TABLE]

where

[TABLE]

and

[TABLE]

In Table 1, we list the number of smooth Schubert varieties of type $\tilde{A}_{n}$, or equivalently, the number of affine permutations in $\tilde{\mathfrak{S}}_{n}$ which avoid $3412$ and $4231$ for $n\leq 9$.

Using the generating series, we can also determine the asymptotics of $a_{n}$. Let

[TABLE]

which is the unique real root of the polynomial $1-6t+8t^{2}-4t^{3}$ from the denominator of the generating function $A(t)$.

Corollary 1.3**.**

*Asymptotically, we have $a_{n}\sim\alpha^{-n}$. *

Proof.

The singularity of $A(t)$ with smallest modulus is the root $\alpha$ of the polynomial $1-t6+8t^{2}-4t^{3}$. Since this occurs with multiplicity one, [FS09, Theorem IV.7] states that $a_{n}\sim C/\alpha^{n+1}$, where $C:=\lim_{t\rightarrow\alpha}A(t)(\alpha-t).$ In this case, $C=\alpha$ (at the moment we do not have an explanation for this interesting coincidence). ∎

In finite type $A$, every rationally smooth Schubert variety is smooth. In affine type $\tilde{A}$, this is not true [BM10, BC12]. For the full flag variety, Billey and Crites show that there is just one infinite family of rationally smooth Schubert varieties which are not smooth.

Theorem 1.4** (Theorem 1.1, Corollary 1.2 and Remark 2.16 of [BC12]).**

A Schubert variety $X(w)$ in the full flag variety of type $\tilde{A}_{n}$ is rationally smooth if and only if either

1. (a)

$w$* avoids the affine permutation patterns $3412$ and $4231$, or* 2. (b)

$w$* is a twisted spiral permutation (in which case, $X(w)$ is not smooth).*

The twisted spiral permutations are defined in the next section. As we will explain, it is easy to see that if $w$ is a twisted spiral permutation, then $X(w)$ is a fibre bundle over a rationally smooth Grassmannian Schubert variety, with fibre equal to the full flag variety of type $A_{n-1}$. The base of this fibre bundle is a spiral Schubert variety, a family of Schubert varieties in the affine Grassmannian introduced by Mitchell [Mit86]. Thus it follows from Theorems 1.1 and 1.4 that, just as for finite-type Schubert varieties, a Schubert variety in the full flag variety of type $\tilde{A}_{n}$ is rationally smooth if and only if it is an iterated fibre bundle of rationally smooth Grassmannian Schubert varieties. As we explain in the next section, this also holds for Schubert varieties in the partial flag varieties of affine type $\tilde{A}_{n}$. Finally, we note that Theorem 1.1 was first proved in a preprint version of [RS16], but was removed during the publication process. Here we give a variant of the original proof, along with an additional proof using staircase diagrams.

The rest of the paper is organized as follows. In the next section, we give the first proof of Theorem 1.1, along with the related results for partial flag varieties. In Section 3, we review the notion of a staircase diagram, and give the second proof of Theorem 1.1. Finally, in Section 4 we prove Theorem 1.2.

1.1. Acknowledgements

We thank Erik Slivken for suggesting the bijection between increasing staircase diagrams and Dyck paths given in the proof of Proposition 4.1. We thank Sara Billey for helpful discussions.

2. BP decompositions in affine type $\tilde{A}$

As in the introduction, let $W$ denote the Weyl group of a Kac-Moody group $G$. Let $S$ be the set of simple reflections of $W$, and let $\ell:W\rightarrow\mathbb{Z}_{\geq 0}$ be the length function. A parabolic subgroup of $W$ is a subgroup $W_{J}$ generated by a subset $J\subseteq S$. Every (left) $W_{J}$-coset has a unique minimal-length element, and the set of minimal-length coset representatives is denoted by $W^{J}$ (similarly, the set of minimal length right coset representatives is denoted by ${}^{J}W$. The partial flag variety $X^{J}$ is stratified by Schubert varieties $X^{J}(w)$ for $w\in W^{J}$, each of complex dimension $\ell(w)$. The Poincaré polynomial of $w\in W^{J}$ is

[TABLE]

where $\leq$ is Bruhat order. As mentioned in the introduction, a theorem of Carrell and Peterson states that $X^{J}(w)$ is rationally smooth if and only if $P^{J}_{w}(q)$ is palindromic, meaning that $q^{\ell(w)}P^{J}_{w}(q^{-1})=P^{J}_{w}(q)$ [Car94]. In the case that $J=\emptyset$, let $X(w):=X^{\emptyset}(w)$ and $P_{w}(q):=P^{\emptyset}_{w}(q)$.

Given $J\subseteq K\subseteq S$, every element $w\in W^{J}$ can be written uniquely as $w=vu$ where $v\in W^{K}$ and $u\in W_{K}^{J}:=W_{K}\cap W^{J}$. This is called the parabolic decomposition of $w$ with respect to $K$. A parabolic decomposition $w=vu$ is a Billey-Postnikov (BP) decomposition (relative to $J$) if $P^{J}_{w}(q)=P^{K}_{v}(q)\cdot P_{u}^{J}(q)$. There are other equivalent combinatorial characterizations which are computationally easy to check. In particular, if $J=\emptyset$, then $w=vu$ is a BP decomposition if and only if $S(v)\cap K\subseteq D_{L}(u)$, where $S(w):=\{s\in S:s\leq w\}$ is the support set of $w\in W$, and $D_{L}(w):=\{s\in S:sw\leq w\}$ is the left descent set of $w$ (see [RS16, Proposition 4.2] for more details).

The following result from [RS16] gives a geometric interpretation for BP decompositions.

Theorem 2.1**.**

([RS16, Theorem 3.3])* Given $J\subseteq K\subset S$, let $w=vu$, where $w\in W^{J}$, $v\in W^{K}$, and $u\in W_{K}^{J}$. Then the following are equivalent:*

1. (a)

The decomposition $w=vu$ is a BP decomposition. 2. (b)

The natural projection $\pi:X^{J}(w)\rightarrow X^{K}(v)$ is Zariski-locally trivial with fibre $X^{J}(u)$.

Consequently, if $w=vu$ is a BP decomposition then $X^{J}(w)$ is (rationally) smooth if and only if $X^{J}(u)$ and $X^{K}(v)$ are (rationally) smooth.

A Grassmannian BP decomposition is a BP decomposition $w=vu$ with respect to a set $K$ with $|S(w)\setminus K|=1$. The main technical result of [BC12] is:

Proposition 2.2** ([BC12]).**

Let $W$ be the Weyl group of type $\tilde{A}_{n}$. If $w\in W$, as an affine permutation avoids $3412$ and $4231$ then either $w$ or $w^{-1}$ has a Grassmannian BP decomposition $vu$, where both $v$ and $u$ belong to proper parabolic subgroups of $\tilde{A}_{n}$.

Proposition 2.2 is proved implicitly in [BC12]; in particular, see the proof of Theorem 3.1, and the discussion before Corollary 7.1 in [BC12]. The following extension of Proposition 2.2 shows that we don’t need to look at $w^{-1}$ to find a BP decomposition:

Proposition 2.3**.**

Let $W$ be a Weyl group of type $\tilde{A}_{n}$. If $w\in W$ avoids both $3412$ and $4231$, then $w$ has a Grassmannian BP decomposition $w=vu$ where both $v$ and $u$ belong to proper parabolic subgroups of $\tilde{A}_{n}$.

Furthermore, one of the following is true:

1. (a)

$w$* is the maximal element of a parabolic proper subgroup of $\tilde{A}_{n}$ or,* 2. (b)

$w=vu$* is a BP decomposition with respect to $S(w)\setminus\{s\}$, for some $s\not\in D_{R}(w)$.*

The proof is similar to that of [RS16, Theorem 6.1]; for the convenience of the reader, we give a complete proof for the $\tilde{A}_{n}$ case.

Proof.

If $S(w)$ is a proper subset of $S$, then $W_{S(w)}$ is finite of type $A$, and the proposition is exactly Theorem 6.1 of [RS16]. Hence, we assume that $S(w)=S$ throughout. By Proposition 2.2, it suffices to prove that if $w^{-1}$ has a Grassmannian BP decomposition with both factors belonging to proper parabolic subgroups, then $w$ has a Grassmannian BP decomposition with respect to some $K=S\setminus\{s\}$ where $s\notin D_{R}(w)$. Assume that $w^{-1}$ has a Grassmannian BP decomposition with both factors belonging to proper parabolic subgroups. Hence there is a subset $K=S\setminus\{s\}$ such that $w=uv$ with $u\in W_{K}$, $v\in{}^{K}W$, and $S(v)\cap K\subseteq D_{R}(u)$, the right descent set of $u$. Furthermore, $S(v)$ is a proper subset of $S=S(w)$; consequently, $S(u)=K$ and $K\setminus S(v)$ is non-empty.

Next, we claim that $u$ has a Grassmannian BP decomposition $u=v^{\prime}u^{\prime}$ with respect to some $K^{\prime}=S\setminus\{s^{\prime}\}$, where $s^{\prime}\not\in S(v)$. Indeed, $u$ is rationally smooth of type $A$. If $u$ is the maximal element of $W_{K}$ then we can take $s^{\prime}$ to be any element of $K\setminus S(v)$. If $u$ is not maximal, then $u$ has a Grassmannian BP decomposition $u=v^{\prime}u^{\prime}$ with respect to $K^{\prime}=K\setminus\{s^{\prime}\}$, where $s^{\prime}\not\in D_{R}(u)$ [RS16, Theorem 6.1], and hence $s^{\prime}\not\in S(v)$.

Since $s^{\prime}\not\in S(v)$, we have $w=v^{\prime}(u^{\prime}v)$ is the parabolic decomposition of $w$ with respect to $K^{\prime}$. Since $v^{\prime}u^{\prime}$ is a BP decomposition, and $u^{\prime}v$ is reduced, we have

[TABLE]

and thus $w=v^{\prime}(u^{\prime}v)$ is a BP decomposition. Finally $S(v^{\prime})\subseteq K\subsetneq S$, completing the proof of the first part of the proposition.

We now show we can choose $s^{\prime}\notin D_{R}(w)$. First, if $u^{\prime}$ is not maximal, then choose a Grassmannian BP decomposition with respect to $s^{\prime}\not\in D_{R}(u)$. Since $s^{\prime}\not\in S(v)$, we get that $s^{\prime}\not\in D_{R}(w)$ as well (this follows, for instance, from the fact that if $a\neq b\in D_{R}(w)$, then $b\in D_{R}(wa)$). If $u^{\prime}$ is instead the maximal element, then we can take $s^{\prime}\in K\setminus S(v)$ such that $s^{\prime}$ is adjacent (in the Dynkin diagram) to some element of $S(v)$. It follows from [RS16, Lemma 6.4] that $s^{\prime}\not\in D_{R}(w)$. ∎

Corollary 2.4**.**

Suppose $w\in W^{J}$, where $W$ is the Weyl group of type $\tilde{A}_{n}$. If $X^{J}(w)$ is a smooth Schubert variety and $S(w)\setminus J\neq\emptyset$, then $w$ has a Grassmannian BP decomposition with respect to some $J\subseteq K\subsetneq S(w)$, in which each factor belongs to a proper parabolic subgroup of $W$.

Proof.

Let $u_{0}$ be the maximal element of $W_{J\cap S(w)}$. Then $w^{\prime}:=wu_{0}$ is a BP decomposition for $w^{\prime}$ with respect to $J$. By Theorem 2.1, $X(w^{\prime})$ is a fibre bundle over $X^{J}(w)$ with fibre $X(u_{0})$. Since $X(u_{0})$ is a full flag variety of type $A$ (and hence is smooth), it follows that $X(w^{\prime})$ is smooth. By Theorem 1.4, $w^{\prime}$ must avoid $3412$ and $4231$. We claim that $w^{\prime}$ has a Grassmannian BP decomposition $w^{\prime}=vu^{\prime}$ with respect to some $K=S(w)\setminus\{s\}$ such that $s\not\in J$. Indeed, if $w^{\prime}$ is the maximal element of $W_{S(w^{\prime})}$, then $w^{\prime}$ has a Grassmannian BP decomposition with respect to any $s\in S(w^{\prime})$, including any element of $S(w)\setminus J$. If $w^{\prime}$ is not maximal, then by Proposition 2.3, $w^{\prime}$ has a Grassmannian BP decomposition where $s\not\in D_{R}(w^{\prime})$, and since $D_{R}(w^{\prime})$ contains $S(u_{0})=J\cap S(w)$, we conclude that $s\not\in J$.

Since $s\not\in J$, we conclude that $K=S\setminus\{s\}$ contains $J$, and thus that $u^{\prime}=uu_{0}$ for some $u\in W_{K}^{J}$. It is easy to see that $w=vu$ is a BP decomposition of $w$ with respect to $K$, as desired. ∎

When combined with Theorem 2.1, Corollary 2.4 implies that every smooth element in type $\tilde{A}$ has a complete BP decomposition in the following sense:

Definition 2.5** ([RS15]).**

Let $W$ be a Coxeter group. A complete BP decomposition of $w\in W^{J}$ (with respect to $J$) is a factorization $w=v_{1}\cdots v_{m}$, where $m=|S(w)\setminus J|$, and, if we let $u_{i}:=v_{i}\cdots v_{m}\in W^{J}$, then $u_{i}=v_{i}u_{i+1}$ is a Grassmannian BP decomposition with respect to $K_{i}:=S(u_{i+1})\cup J$ for all $1\leq i<m$.

A complete maximal BP decomposition $w=v_{1}\cdots v_{m}$ is a complete BP decomposition $w=v_{1}\cdots v_{m}$ as above such that $v_{i}$ is maximal in $W_{K_{i-1}\cap S(v_{i})}^{S(v_{i})}$ for all $1\leq i\leq m$.

Hence (and similarly to [RS16, Corollary 3.7]), Theorem 2.1 and Corollary 2.4 imply that a Schubert variety $X^{J}(w)$ of type $\tilde{A}_{n}$ is smooth if and only if it is an iterated fibre bundle of Grassmannians.

Corollary 2.6**.**

A Schubert variety $X^{J}(w)$ in a partial flag variety of type $\tilde{A}_{n}$ is smooth if and only if there is a sequence

[TABLE]

where each map $X_{i}\rightarrow X_{i-1}$ is a Zariski-locally trivial fibre bundle whose fibre is a Grassmannian variety of type $A$.

Proof.

A Zariski-locally trivial fibre bundle with a smooth base and fibre is itself smooth, so it follows that if $X^{J}(w)$ is an iterated fibre bundle in the above sense, then $X^{J}(w)$ is smooth.

For the converse, let $W$ be the Weyl group of type $\tilde{A}_{n}$, and suppose that $X^{J}(w)$ is smooth, where $w\in W^{J}$. If $S(w)\subseteq J$, then $X^{J}(w)$ is a point, and the theorem is trivial. If $|S(w)\setminus J|=m\geq 1$, then by Corollary 2.4, $w$ has a complete BP decomposition $w=v_{1}\cdots v_{m}$ in which $S(v_{i})$ is a strict subset of $S$. Let $u_{i}:=v_{i}\cdots v_{m}\in W^{J}$, and let $w_{i}:=v_{1}\cdots v_{i}$, so that $u_{1}=w_{m}=w$ and $u_{m+1}=w_{0}=e$. In addition, set $K_{i}:=S(u_{i+1})\cup J$, so that

• •

$J=K_{m}\subsetneq K_{m-1}\subsetneq\cdots\subsetneq K_{1}\subsetneq K_{0}:=S(w)$,

• •

$|K_{i-1}\setminus K_{i}|=1$,

• •

$v_{i}\in W_{K_{i-1}}^{K_{i}}$,

• •

$u_{i}=v_{i}u_{i+1}$ is a BP decomposition with respect to $K_{i}$ (and relative to $J$).

By Theorem 2.1, $X^{K_{i}}(v_{i})$ is smooth for all $1\leq i\leq m$. Since $S(v_{i})$ is a strict subset of $S$, it follows that $X^{K_{i}}(v_{i})$ is a smooth Grassmannian Schubert variety of finite type $A$. But it is well-known (see for instance [BL00]) that the only smooth Grassmannian Schubert varieties of finite type $A$ are themselves Grassmannians (specifically, $v_{i}$ must be maximal in $W_{S(v_{i})}^{K_{i}\cap S(v_{i})}$, so $w=v_{1}\cdots v_{m}$ is a complete maximal BP decomposition). By [RS16, Lemma 4.3], $w_{i}=w_{i-1}v_{i}$ is a BP decomposition with respect to $K_{i-1}$ (and relative to $K_{i}$), so if we set $X_{i}=X^{K_{i}}(w_{i})$, then by Theorem 2.1 the standard projection $X_{i}\rightarrow X_{i-1}$ is a Zariski-locally trivial fibre bundle with fibre $X^{K_{i}}(v_{i})$. ∎

Proof of Theorem 1.1.

If $X(w)$ is smooth, then $w$ avoids $3412$ and $4231$ by Theorem 1.4, so part (a) implies part (b).

Suppose $w$ avoids $3412$ and $4231$. If $w=vu$ is a parabolic decomposition, then $u$ also avoids $3412$ and $4231$ [BC12, Lemma 3.10]. Thus Proposition 2.3 implies that $w$ has a complete BP decomposition, in which every factor belongs to a proper parabolic subgroup of $W$. Every rationally smooth Grassmannian Schubert variety in finite type $A$ is smooth by the Carrell-Peterson theorem [CK03], and hence a Grassmannian. Thus the proof of Corollary 2.6 implies that $X(w)$ is an iterated fibre bundle of Grassmannians. Hence part (b) implies part (c).

Similarly, if $X(w)$ is an iterated fibre bundle of Grassmannians, then $X(w)$ is smooth as in the proof of Corollary 2.6, so part (c) implies part (a). ∎

We finish the section by looking at rationally smooth Schubert varieties, starting with the twisted spiral permutations. Suppose $W$ has type $\tilde{A}_{n}$, and let $S=\{s_{0},\ldots,s_{n-1}\}$, where $s_{i}$ is the simple reflection corresponding to node $i$ in the Dynkin diagram of $\tilde{A}_{n}$ as shown in Figure 1.

Given $0\leq i<n$ and $k\geq 1$, define

[TABLE]

where the indices of the $s_{j}$’s are interpreted modulo $n$. Both $x(i,k)$ and $y(i,k)$ belong to $W^{S\setminus\{s_{i}\}}$. A spiral permutation is an element of $W$ of the form $x(i,k(n-1))$ or $y(i,k(n-1))$ for some $k\geq 2$.222Note that the length of these elements is a multiple of $n-1$, even though the rank of $W$ is $n$. In particular, if $k=1$ then these elements belong to a parabolic subgroup of finite type $A$, so we exclude this case. The spiral permutations were first studied by Mitchell [Mit86], who showed that the corresponding Grassmannian Schubert varieties, called spiral Schubert varieties, are rationally smooth.

A twisted spiral permutation is an element of $W$ of the form $w=vu$, where $v$ is a spiral permutation $x(i,k(n-1))$ or $y(i,k(n-1))$, and $u$ is the maximal element of $W_{S\setminus\{s_{i}\}}$. Note that $w=vu$ is a BP decomposition with respect to $J=S\setminus\{s_{i}\}$. In addition, $D_{R}(w)$ contains $D_{R}(u)=S\setminus\{s_{i}\}$, and since $\tilde{A}_{n}$ is infinite, $D_{R}(w)$ cannot be equal to $S$, so $D_{R}(w)$ must be equal to $S\setminus\{s_{i}\}$. Thus we get a version of Proposition 2.3 for all rationally smooth elements of $W$: if $X(w)$ is rationally smooth, then either

• •

$w$ is the maximal element of a proper parabolic subgroup of $W$, or

• •

$w$ has a Grassmannian BP decomposition with respect to $K=S(w)\setminus\{s\}$, where $s\not\in D_{R}(w)$.

This means that we can repeat the proofs of Corollaries 2.4 and 2.6 to get:

Corollary 2.7**.**

A Schubert variety $X^{J}(w)$ in a partial flag variety of type $\tilde{A}_{n}$ is rationally smooth if and only if there is a sequence

[TABLE]

where each map $X_{i}\rightarrow X_{i-1}$ is a Zariski-locally trivial fibre bundle whose fibre is a rationally smooth Grassmannian Schubert variety of type $A$ or $\tilde{A}$.

It is implicit in the proof of Corollary 2.7 that if $X^{J}(w)$ is rationally smooth, then we can construct such a sequence where all the fibres are either Grassmannians of finite type $A$, or spiral Schubert varieties. In fact, these are the only rationally smooth Grassmannian Schubert varieties, by a theorem of Billey and Mitchell [BM10].

In the finite-type analogue of Corollary 2.7, every rationally smooth Grassmannian Schubert variety is almost-maximal [RS16]. Although we don’t need that fact here, it is interesting to note that the spiral permutations are also almost-maximal.

3. Staircase diagrams for affine type $\tilde{A}$

The main fact we had to establish in the previous section was that every 3412- and 4231-avoiding element of type $\tilde{A}$ has a complete maximal BP decomposition. To prove this fact, we showed that the existence of BP decompositions for $w$ or $w^{-1}$ implies the existence of BP decompositions for $w$. The same proof strategy was used in [RS16]. Both results are instances of a more general result, which we formulate as follows:

Proposition 3.1**.**

Let $\mathcal{F}$ be a family of Coxeter groups which is closed under parabolic subgroups (i.e. if $W\in\mathcal{F}$, then $W_{J}\in\mathcal{F}$ for every $J\subseteq S$). Let $\mathcal{C}$ be a class of elements of the groups of $\mathcal{F}$ such that if $w\in\mathcal{C}$, then

1. (1)

$w^{-1}\in\mathcal{C}$, and 2. (2)

$u\in\mathcal{C}$* for all parabolic decompositions $w=vu$.* 3. (3)

either $w$ or $w^{-1}$ has a maximal Grassmannian BP decomposition, i.e. a Grassmannian BP decomposition $vu$ where $v$ is maximal in $W_{S(v)}^{S(u)\cap S(v)}$.

Then every $w\in\mathcal{C}$ has a complete maximal BP decomposition.

The point of this section is to show that Proposition 3.1 has a short proof using staircase diagrams. Taking $\mathcal{F}$ to be the Weyl groups of finite type $A$ and affine type $\tilde{A}$, and $\mathcal{C}$ to be the class of permutations avoiding the pattern $3412$ and $4231$, we get a proof of Theorem 1.1 by a somewhat different route. Our proof of Proposition 3.1 will still hold if we replace “maximal” by “maximal or almost-maximal” BP decompositions, and hence Proposition 3.1 also gives an alternate path to the results on existence of BP decompositions in [RS16]. For simplicity, we restrict ourselves to maximal BP decompositions.

As we will also use staircase diagrams in the next section, we briefly review the definition from [RS15]. Let $G$ be a graph with vertex set $S$, and recall that a subset $B\subseteq S$ is connected if the subgraph of $G$ induced by $B$ is connected. If $\mathcal{D}$ is a collection of subsets of $S$ and $s\in S$, we set

[TABLE]

Staircase diagrams are then defined as follows:

Definition 3.2** ([RS15]).**

Let $\Gamma$ be a graph with vertex set $S$. Let $\mathcal{D}=(\mathcal{D},\preceq)$ be a partially ordered subset of $2^{S}$ not containing the empty set. We say that $\mathcal{D}$ is a staircase diagram if the following are true:

1. (1)

Every $B\in\mathcal{D}$ is connected, and if $B$ covers $B^{\prime}$ then $B\cup B^{\prime}$ is connected. 2. (2)

The subset $\mathcal{D}_{s}$ is a chain for every $s\in S$. 3. (3)

If $s\operatorname{adj}t$, then $\mathcal{D}_{s}\cup\mathcal{D}_{t}$ is a chain, and $\mathcal{D}_{s}$ and $\mathcal{D}_{t}$ are saturated subchains of $\mathcal{D}_{s}\cup\mathcal{D}_{t}$. 4. (4)

If $B\in\mathcal{D}$, then there is some $s\in S$ (resp. $s^{\prime}\in S$) such that $B$ is the minimum element of $\mathcal{D}_{s}$ (resp. maximum element of $\mathcal{D}_{s^{\prime}}$).

The definition is symmetric with respect to the partial order, so if $\mathcal{D}=(\mathcal{D},\preceq)$ is a staircase diagram, and $\preceq^{\prime}$ is the reverse order to $\preceq$, then $(\mathcal{D},\preceq^{\prime})$ is also a staircase diagram, called $\operatorname{flip}(\mathcal{D})$. Finally, if $\mathcal{D}$ is a staircase diagram over the Coxeter graph of a Coxeter group $W$, we say $\mathcal{D}$ is spherical if $W_{B}$ is a finite group for all $B\in\mathcal{D}$. The main result about staircase diagrams is:

Theorem 3.3** ([RS15], Theorem 5.1, Theorem 3.7, and Corollary 6.4).**

Let $\Gamma$ be the Coxeter graph of a Weyl group $W$. Then there is a bijection between spherical staircase diagrams over $\Gamma$, and elements of $w$ with a complete maximal BP decomposition (relative to $\emptyset$).

Furthermore, if a staircase diagram $\mathcal{D}$ corresponds to $w\in W$, then $\operatorname{flip}(\mathcal{D})$ corresponds to $w^{-1}$.

Second proof of Proposition 2.3.

The proof is by induction on $|S(w)|$. Suppose that $w$ has a maximal Grassmannian BP decomposition $w=vu$. Since $|S(u)|<|S(v)|$, we can conclude by induction that $u$ has a complete maximal BP decomposition. But these means that $w$ has a complete maximal BP decomposition, and we are done. Similarly, if $w^{-1}$ has a maximal Grassmannian BP decomposition, then we conclude that $w^{-1}$ has a complete maximal BP decomposition. But this means that $w^{-1}$ comes from a staircase diagram $\mathcal{D}$, so $w$ corresponds to $\operatorname{flip}(\mathcal{D})$. In particular, $w$ must also have a complete maximal BP decomposition. ∎

4. Enumeration of smooth Schubert varieties

Let $\tilde{\Gamma}_{n}$ be the Coxeter graph of type $\tilde{A}_{n}$ with vertices $\tilde{S}_{n}=\{s_{0},s_{1},\ldots,s_{n-1}\}$ as in Figure 1. In this case, proper, connected subsets of $\tilde{S}_{n}$ are simply intervals on the cycle graph $\tilde{\Gamma}_{n}$. Define the interval

[TABLE]

We represent a staircase diagram $\mathcal{D}$ pictorially with a collection of “blocks” where if $B_{2}$ lies above $B_{1}$ and $B_{1}\cup B_{2}$ is connected, then $B_{1}\prec B_{2}$ in $\mathcal{D}$. For example, the staircase diagram

[TABLE]

over $\tilde{\Gamma}_{10}$ could be represented in Figure 2.

However, for the sake of convenience we will represent the cycle graph $\tilde{\Gamma}_{n}$ as a line graph with vertex $s_{0}$ each end point, as follows.

[TABLE]

We can then draw the staircase diagram in two-dimensions; for instance, the staircase diagram in Figure 2 is represented as:

[TABLE]

Note that if $s_{0}\in B\in\mathcal{D}$, then $B$ appears as a “disconnected” block in the pictorial representation of $\mathcal{D}$.

By Corollary 2.6 and Theorem 3.3, to enumerate smooth Schubert varieties of type $\tilde{A}_{n}$, it suffices to enumerate spherical staircase diagrams over the graph $\tilde{\Gamma}_{n}$. Since every proper subgraph of $\tilde{\Gamma}_{n}$ is a Dynkin diagram of finite type, the only non-spherical staircase diagram is $\mathcal{D}=\{\tilde{S}_{n}\}$.

We first consider staircase diagrams over the Dynkin graph of finite type $A_{n}$. Let $\Gamma_{n}$ denote the line graph with vertex set $S_{n}=\{s_{1},\ldots,s_{n}\}$. In this case, we denote interval blocks on line graph $\Gamma_{n}$ as simply $[s_{i},s_{j}]$ for $i<j$.

[TABLE]

A staircase diagram $\mathcal{D}$ is said to have full support if every vertex of the underlying graph appears in some $B\in\mathcal{D}$. We say a staircase diagram $\mathcal{D}$ over $\Gamma_{n}$ of full support is increasing if we can write

[TABLE]

with $s_{1}\in B_{1}$. Similarly, a staircase diagram is decreasing if if we can write $\mathcal{D}=\{B_{1}\succ B_{2}\succ\cdots\succ B_{m}\}$ with $s_{1}\in B_{1}$. Let $M^{\pm}(n)$ denote the set of fully supported increasing/decreasing staircase diagrams over $\Gamma_{n}$. Define the generating series

[TABLE]

where $m_{n}:=|M^{+}(n)|=|M^{-}(n)|$.

Proposition 4.1**.**

The generating function $\displaystyle A_{M}(t)=\frac{1-2t-\sqrt{1-4t}}{2t}$.

Proof.

The proposition can be proved by modifying the proof of [RS15, Proposition 8.3]. In this paper we present an alternate proof by giving a bijection between fully supported increasing staircase diagrams and Dyck paths. Indeed, let $\mathcal{D}=\{B_{1}\prec B_{2}\prec\cdots\prec B_{m}\}$ be a fully supported increasing staircase diagram on $\Gamma_{n}$. For each $B_{i}\in\mathcal{D},$ define the numbers

[TABLE]

where we set $B_{0}=B_{m+1}=\emptyset.$ Let $P(\mathcal{D})$ denote the lattice path in $\mathbb{Z}^{2}$ from $(0,0)$ to $(n,n)$ which takes $r(B_{1})$ steps to the right, then $u(B_{1})$ steps going up, followed by $r(B_{2})$ steps to the right, then $u(B_{2})$ steps going up and so forth (See Example 4.2). Since $\mathcal{D}$ is fully supported, we have that

[TABLE]

and hence $P(\mathcal{D})$ terminates at $(n,n)$. Definition 3.2 implies $r(B_{i}),u(B_{i})>0$ and that the partial sums

[TABLE]

for all $i\leq m.$ Thus $P(\mathcal{D})$ is a Dyck path. Conversely, any Dyck path is given by a sequence of positive pairs $(r_{i},u_{i})$ giving steps to the right followed up steps going up. Set $u_{0}:=0$ and define

[TABLE]

and $\bar{\mathcal{D}}:=\{\bar{B}_{1}\prec\bar{B}_{2}\prec\cdots\prec\bar{B}_{m}\}$. It is easy to see that $\bar{\mathcal{D}}$ is a fully supported staircase diagram and that this construction is simply the inverse of the map $P$. The proposition now follows from the generating function for Dyck paths which is given by Catalan numbers. ∎

Example 4.2**.**

Consider the staircase diagram $\mathcal{D}=({s_{1}}\prec[s_{2},s_{5}]\prec[s_{4},s_{6}])$ on $\Gamma_{6}$. The sequence of pairs $(r_{i},u_{i})$ is $((1,1),(4,2),(1,3))$ and corresponding Dyck path $P(\mathcal{D})$ is given below.

[TABLE]

The idea behind enumerating staircase diagrams over $\tilde{\Gamma}_{n}$ is to partition a staircase diagram into a disjoint union of increasing and decreasing staircase diagrams of finite type $A$. To do this precisely, we introduce the notion of a broken staircase diagram. We say that a partially ordered collection of subsets $(\mathcal{B},\prec)$ of vertices of the graph $\Gamma_{n}$ is a broken staircase diagram if

[TABLE]

for some $\mathcal{D}\in M^{+}(n+1)\cup M^{-}(n+1)$ where the partial order on $\mathcal{B}$ is induced from $\mathcal{D}$. Note that broken staircase diagrams are allowed to violate part (4) of Definition 3.2, and must be either increasing or decreasing. In particular, if $\mathcal{B}=\{B_{1}\prec\cdots\prec B_{m}\}$ is broken, then it may be possible for $B_{m}\subset B_{m-1}$. Define the generating series

[TABLE]

where $b_{n}$ denotes the number of increasing (or equivalently, decreasing) broken staircase diagrams on $\Gamma_{n}$.

Proposition 4.3**.**

The generating function $\displaystyle A_{B}(t)=\frac{(1-t)A_{M}(t)}{t}-1.$

Proof.

Clearly $b_{0}=0$ and $b_{1}=1$, so we assume that $n\geq 2$. Let $\mathcal{D}=\{B_{1}\prec\cdots\prec B_{m}\}\in M^{+}(n+1)$ and let $\mathcal{B}(\mathcal{D})=\{B\cap S_{n}\ |\ B\in\mathcal{D}\}$ denote the corresponding broken staircase diagram. If block index $k\leq m-2$, then $B_{k}\subseteq S_{n}$. Hence $\mathcal{B}(\mathcal{D})$ determines $\mathcal{D}$ up to the last two blocks $B_{m-1},\,B_{m}$. If $B_{m}\subset B_{m-1}$, then $\mathcal{B}(\mathcal{D})$ uniquely determines $\mathcal{D}$ as shown in Figure 3.

If $B_{m}\not\subset B_{m-1}$, then there are two possibilities for $\mathcal{D}$ given $\mathcal{B}(\mathcal{D})$. Either $\mathcal{D}_{s_{n}}=\{B_{m}\}$ and thus $s_{n},s_{n+1}\in B_{m}$, or $\mathcal{D}_{s_{n}}=\{B_{m-1}\}$ which implies $B_{m}=\{s_{n+1}\}$ (see Figure 4).

In the latter case, removing the last block from $\mathcal{D}$ gives a unique staircase diagram in $M^{+}(n)$. Hence $b_{n}=m_{n+1}-m_{n}$ and

[TABLE]

This proves the proposition. ∎

We can now state the main bijection:

Proposition 4.4**.**

There is a bijection between fully-supported spherical staircase diagrams on $\tilde{\Gamma}_{n}$, and pairs $[(\mathcal{B}_{1},\ldots,\mathcal{B}_{2k}),v]$, where

• •

$\mathcal{B}_{i}$* is a broken staircase on $\Gamma_{n_{i}}$, $n_{i}\geq 1$,*

• •

$\sum_{i=1}^{2k}n_{i}=n$,

• •

for all $1\leq i\leq 2k-1$, if $\mathcal{B}_{i}$ is increasing (resp. decreasing) then $\mathcal{B}_{i+1}$ is decreasing (resp. increasing), and

• •

$v$* is a distinguished vertex in $\mathcal{B}_{2k}$.*

Proof.

For the purpose of this proof, we let $s_{j+nk}=s_{j}$ for any $k$ and $0\leq j<n$. Suppose $\mathcal{D}$ is a fully-supported staircase diagram on $\tilde{\Gamma}_{n}$, and $B$ is any block of $\mathcal{D}$, say $B=\overrightarrow{[s_{i},s_{j}]}$. By Definition 3.2, part (3), all the blocks of $\mathcal{D}_{s_{j+1}}$ are comparable with $B$. If $B$ has an upper cover $B^{\prime}\succeq B$ in $\mathcal{D}_{s_{j}}\cup\mathcal{D}_{s_{j+1}}$, then there are no elements of $\mathcal{D}_{s_{j+1}}$ below $B$, since then $\mathcal{D}_{s_{j+1}}$ would not be saturated in $\mathcal{D}_{s_{j}}\cup\mathcal{D}_{s_{j+1}}$. And vice-versa, if $B$ has a lower cover in $\mathcal{D}_{s_{j}}\cup\mathcal{D}_{s_{j+1}}$ then there are no elements of $\mathcal{D}_{s_{j+1}}$ above $B$. Consequently we can say that $B$ has a unique cover $B^{\prime}$ containing $s_{j+1}$. We call $B^{\prime}$ the right cover of $B$.

Choose some block $B_{1}$, and let $B_{1},\ldots,B_{m}$ be a sequence where $B_{i+1}$ is the right cover of $B_{i}$ for $1\leq i<m$, and $B_{1}$ is the right cover of $B_{m}$. Then every block of $\mathcal{D}$ must appear in this sequence. Indeed, every vertex of $\tilde{\Gamma}_{n}$ appears in some block in this sequence, so every block of $\mathcal{D}$ is comparable to some element of the sequence. It follows that if there is a block of $\mathcal{D}$ not in the sequence, then there is a block $B$ not in the sequence which has an upper or lower cover $B^{\prime}$ in the sequence. Then either $B$ will be the right cover of $B^{\prime}$, or $B^{\prime}$ will be the right cover of $B$. But the same argument as above shows that $B^{\prime}$ has a unique left cover, and this is the only element with $B^{\prime}$ as a right cover. So in both cases, $B$ must also be in the sequence, a contradiction.

Let $B_{i_{1}},\ldots,B_{i_{m}}$ denote the subsequence of extremal blocks, i.e. blocks which are maximal or minimal. Note that if $B_{i_{j}}$ is maximal then $B_{i_{j+1}}$ must be minimal, and vice-versa. Since $\tilde{\Gamma}_{n}$ is a cycle, the same must apply to $B_{i_{m}}$ and $B_{i_{1}}$, and in particular $m$ must be even. By Definition 3.2, part (4), every extremal block contains a vertex which does not belong to any block. Let $1\leq c_{j}\leq n$ be the index of the leftmost such vertex in $B_{i_{j}}$. By cyclically shifting the indices, we can assume that $1\leq c_{1}<c_{2}<\ldots<c_{m}<n$. Finally, set

[TABLE]

so that $J_{1},\ldots,J_{m}$ partitions $\tilde{S}_{n}$, and let

[TABLE]

with the induced partial order. Since $B_{i_{j}}$ is the only block containing $s_{c_{j}}$, and no block of $\mathcal{D}$ contains any other [RS15, Lemma 2.6(b)], the block $B_{i_{j}}$ can meet at most two of the intervals $J_{k}$. Hence $\mathcal{B}_{j}$ is either an increasing or decreasing chain. It follows that $\mathcal{B}_{1},\ldots,\mathcal{B}_{m}$ is a sequence of broken staircases as required. We set $v$ to be the vertex $s_{n-1}$, which is always in $\mathcal{B}_{m}$ by construction.

This construction gives a map from staircase diagrams to sequences of broken staircases with a marked vertex. To show that this map has an inverse, suppose that $\mathcal{B}=\{B_{1}\prec\cdots\prec B_{m}\}$ is an increasing broken staircase. If $B_{m}\subset B_{m-1}$ then we can think of $\mathcal{B}$ as the staircase diagram $\{B_{1}\prec\cdots\prec B_{m-1}\}$ with an additional broken block $B_{m}$ on top of $B_{m-1}$. If $B_{m}\not\subset B_{m-1}$, so $\mathcal{B}$ is a staircase diagram in its own right, then we think of $\mathcal{B}$ as a staircase diagram with an empty broken block on top of the block $B_{m}$, starting and ending after the rightmost vertex of $B_{m}$. If $\mathcal{B}^{\prime}$ is then a decreasing staircase diagram, we can glue $\mathcal{B}$ and $\mathcal{B}^{\prime}$ together by attaching the broken block of $\mathcal{B}$ to the first block of $\mathcal{B}^{\prime}$. We can similarly glue a decreasing broken staircase to an increasing broken staircase. Given a sequence $\mathcal{B}_{1},\ldots,\mathcal{B}_{2k}$ of alternately increasing and decreasing broken staircases, we can glue them together in order, and then glue $\mathcal{B}_{2k}$ to $\mathcal{B}_{1}$ to get a staircase diagram on a cycle. Labelling the vertices of the cycle with $s_{0},\ldots,s_{n-1}$ starting to the right of the marked vertex $v$, we get a staircase diagram on $\tilde{\Gamma}_{n}$, and this process inverts the above map. ∎

Example 4.5**.**

The staircase diagram

[TABLE]

has four extremal blocks and partitions into an alternating sequence of increasing and decreasing broken staircase diagrams as follows:

[TABLE]

Define the generating series

[TABLE]

where $\bar{a}_{n}$ denotes the number of fully supported spherical staircase diagrams on $\tilde{\Gamma}_{n}.$

Corollary 4.6**.**

The generating function $\displaystyle\bar{A}(t)=\frac{2A_{B}(t)\cdot t\frac{d}{dt}A_{B}(t)}{1-A_{B}(t)^{2}}.$

Proof.

Follows immediately from Proposition 4.4, and the fact that $t\frac{d}{dt}A_{B}(t)$ is the generating series for broken staircases with a marked vertex. Note that we get a factor of two because the first broken staircase can be increasing for decreasing. ∎

If a staircase diagram on $\tilde{\Gamma}_{n}$ is not fully supported, then it is a disjoint union of fully supported staircase diagrams over a collection of subpaths of the cycle. Let $f_{n}$ denote the number of fully supported staircase diagrams on the path $\Gamma_{n}$ and define the generating series

[TABLE]

The following proposition is proved in [RS15, Proposition 8.3]. We give an alternate proof using broken staircase diagrams.

Proposition 4.7**.**

([RS15, Proposition 8.3])* The generating function $\displaystyle A_{F}(t)=\frac{A_{M}(t)}{1-A_{B}(t)}$.*

Proof.

We emulate the proof of Proposition 4.4 as follows: Given a staircase diagram $\mathcal{D}$ on $\Gamma_{n}$, let $B_{1},\ldots,B_{k}$ be the maximal and minimal blocks in order from left to right. Let $1\leq m\leq k$ be the largest index such that $B_{m}$ is minimal in $\mathcal{D}$ (so actually, $m\in\{k-1,k\}$). For every $1\leq j\leq m$, let $a_{j}$ be the index of the leftmost element of $B_{j}$ which is not contained in any other block, and let

[TABLE]

Let $\mathcal{B}_{i}=[s_{a_{i}},s_{b_{i}}]$. Then $\mathcal{B}_{i}$ is a broken staircase for $1\leq i\leq m-1$, while $\mathcal{B}_{m}$ is an increasing staircase. This also implies that $\mathcal{B}_{m-1}$ is decreasing, $\mathcal{B}_{m-2}$ is increasing, and so on. It is not hard to see that this map is a bijection, and hence every staircase diagram over $\Gamma_{n}$ decomposes into a sequence of broken staircases, followed by an increasing staircase. ∎

One subtlety of the above bijection is that it seems to miss the case when $\mathcal{D}$ is decreasing, or more generally, ends with a decreasing staircase. However, a decreasing staircase decomposes into a decreasing broken staircase followed by a single block. Since a single block is an increasing staircase, the bijection will in fact count decreasing staircases correctly. Since single blocks are both increasing and decreasing, the seemingly more straightforward approach of allowing $\mathcal{B}_{m}$ to be increasing or decreasing will lead to overcounts. The reason this problem doesn’t arise in Proposition 4.4 is that in that bijection every $\mathcal{B}_{i}$ is a broken staircase. We can always tell whether a broken staircase is increasing or decreasing based on where it is glued to the adjacent broken staircase.

Finally, let $a_{n}$ denote the number of spherical staircase diagrams over the graph $\tilde{\Gamma}_{n}$ and define

[TABLE]

Proposition 4.8**.**

The generating series

[TABLE]

where $\displaystyle A_{*}(t)=\frac{tA_{F}(t)}{1-t}.$

Proof.

First note that the generating function fully supported staircase diagrams over $\tilde{\Gamma}_{n}$ is $\bar{A}(t)$. If a staircase diagram is not fully supported and nonempty, then it partitions into a sequence $(\mathcal{D}_{0},\mathcal{E}_{0},\ldots,\mathcal{D}_{r},\mathcal{E}_{r})$ where each $\mathcal{D}_{k}$ is a nonempty, fully supported staircase diagram over a path and $\mathcal{E}_{k}$ is an empty staircase diagram over a path of at least length one. Moreover, we can choose such a partition such that $s_{0}$ is in the support of $\mathcal{D}_{0}$ or $\mathcal{E}_{0}$. Thus we get a bijection between non-empty non-fully-supported staircase diagrams on $\tilde{\Gamma}_{n}$ and sequences $(\mathcal{D}_{0},\mathcal{E}_{0},\ldots,\mathcal{D}_{r},\mathcal{E}_{r})$ where $(\mathcal{D}_{0},\mathcal{E}_{0})$ has a marked vertex corresponding to $s_{0}$. The generating series for staircase diagrams corresponding to pairs $(\mathcal{D}_{k},\mathcal{E}_{k})$ over $\Gamma_{n}$ is

[TABLE]

Thus the generating function for the number non-fully support staircase diagrams over $\tilde{\Gamma}_{n}$ is

[TABLE]

where the second summand corresponds to the generating function of empty staircase diagrams. This completes the proof. ∎

Proof of Theorem 1.2.

Combine Propositions 4.1, 4.3, 4.7, and 4.8, along with Corollary 4.6. ∎