Bivariate Order Polynomials

TL;DR

This paper introduces a bivariate order polynomial extending Stanley's order polynomial, with decomposition formulas, reciprocity theorems, and links to bivariate chromatic polynomials, inspired by graph coloring concepts.

Contribution

It presents a novel bivariate order polynomial with decomposition, reciprocity, and graph coloring connections, expanding combinatorial polynomial theory.

Findings

Derived decomposition formulas in terms of linear extensions

Established a combinatorial reciprocity theorem

Connected the new polynomial to bivariate chromatic polynomials

Abstract

Motivated by Dohmen-P\"onitz-Tittmann's bivariate chromatic polynomial $\chi_G(x,y)$, which counts all $x$-colorings of a graph $G$ such that adjacent vertices get different colors if they are $\le y$, we introduce a bivarate version of Stanley's order polynomial, which counts order preserving maps from a given poset to a chain. Our results include decomposition formulas in terms of linear extensions, a combinatorial reciprocity theorem, and connections to bivariate chromatic polynomials.

Figures

Equations

$$c(v)\ne c(w)\text{ or }c(v)=c(w)>y.$$

$$a⪯b⟹\phi (a)\le \phi (b).$$

$$a\prec b⟹\phi (a)<\phi (b).$$

$${\chi }_G(x)=\sum _{\sigma \text{ acyclic orientation of }G}{\Omega }_{\sigma }^{\circ }(x).$$

$$a⪯b⟹\phi (a)\le \phi (b)\text{ for all }a,b\in P\text{ and }\phi (c)\ge y\text{ for all }c\in C.$$

$$a\prec b⟹\phi (a)<\phi (b)\text{ for all }a,b\in P\text{ and }\phi (c)>y\text{ for all }c\in C.$$

$$1\le \phi (a_1)\le y<\phi (a_2)\le x$$

$$1\le y<\phi (a_1)<\phi (a_2)\le x$$

$$(-1{)}^{|P|}{\Omega }_{P,C}^{\circ }(-x,-y)={\Omega }_{P,C}(x,y+1).$$

$${\chi }_G(-x,-y)=\sum _{H\text{ flat of }G}(-1{)}^{|V(H)|}{m}_H(\sigma ,c),$$

$${\Omega }_{P,C}^{\circ }(x,y)=\sum _{i=0}^k\left(\genfrac{}{}{0ex}{}{y}{i}\right)\left(\genfrac{}{}{0ex}{}{x-y}{n-i}\right)\text{ and }{\Omega }_P(x,y)=\sum _{i=0}^k\left(\genfrac{}{}{0ex}{}{y-2+i}{i}\right)\left(\genfrac{}{}{0ex}{}{x-y+n-i}{n-i}\right).$$

$$1\le \phi (a_1)<\cdots <\phi (a_k)\le y<\phi (a_{k+1})<\cdots <\phi (a_n)\le x$$

$$1\le \phi (a_1)<\cdots <\phi (a_{k-1})\le y<\phi (a_k)<\cdots <\phi (a_n)\le x$$

$$1\le y<\phi (a_1)<\cdots <\phi (a_n)\le x$$

$$(-1{)}^n{\Omega }_{P,C}^{\circ }(-x,-y)={\Omega }_{P,C}(x,y+1).$$

$$\displaystyle\Omega^{\circ}_{P,\,C}(-x,-y)\$$

$$=\sum _{i=0}^k(-1{)}^i(\genfrac{}{}{0ex}{}{y+i-1}{i})(-1{)}^{n-i}\left(\genfrac{}{}{0ex}{}{x-y+n-i-1}{n-i}\right)$$

$$=(-1{)}^n{\Omega }_{P,C}(x,y+1).\text{qed}$$

$$a{⪯}_{P}b⟹a{⪯}_{L}b.$$

$$\mathrm{Asc}({\omega }_L):=\left\{j:\omega (a_j)<\omega (a_{j+1})\right\}\text{ and }\mathrm{Des}({\omega }_L):=\left\{j:\omega (a_j)>\omega (a_{j+1})\right\}$$

$$1\le \phi (a)\le \phi (d)<\phi (b)\le \phi (c)\le \phi (e)\le x.$$

$$\bar{\phi }(a)$$

$$\bar{\phi }(d)$$

$$\bar{\phi }(b)$$

$$\bar{\phi }(c)$$

$$\bar{\phi }(e)$$

$$1\le \bar{\phi }(a)<\bar{\phi }(d)<\bar{\phi }(b)<\bar{\phi }(c)<\bar{\phi }(e)\le x+3.$$

$${\Omega }_{14235}^{\circ }(x,y)=\sum _{i=0}^2\left(\genfrac{}{}{0ex}{}{y-2}{i}\right)\left(\genfrac{}{}{0ex}{}{x-3-y+2}{5-i}\right).$$

$${\Omega }_{{\omega }_L}^{\circ }(x,y)=\sum _{i=0}^k\left(\genfrac{}{}{0ex}{}{y-\mathrm{asc}({\tilde{\omega }}_L)}{i}\right)\left(\genfrac{}{}{0ex}{}{x-y+\mathrm{asc}({\tilde{\omega }}_L)-\mathrm{asc}({\omega }_L)}{n-i}\right)$$

$${\Omega }_{{\omega }_L}(x,y)=\sum _{i=0}^k\left(\genfrac{}{}{0ex}{}{y-\mathrm{des}({\tilde{\omega }}_L)-2+i}{i}\right)\left(\genfrac{}{}{0ex}{}{x-y+\mathrm{des}({\tilde{\omega }}_L)-\mathrm{des}({\omega }_L)+n-i}{n-i}\right).$$

$$\bar{\phi }(a_i)=\phi (a_i)+\mathrm{asc}\left(\omega (a_1)\cdots \omega (a_i)\right)\text{for }2\le i\le n.$$

$$\phi (a_i)<\phi (a_{i+1})⟹\bar{\phi }(a_i)<\bar{\phi }(a_{i+1}).$$

$$\phi (a_i)\le \phi (a_{i+1})⟹\bar{\phi }(a_i)<\bar{\phi }(a_{i+1}).$$

Full text

1. Introduction

Graph coloring problems are ubiquitous in many areas within and outside of mathematics. The motivation of our study is the bivariate chromatic polynomial $\chi_{G}(x,y)$ of a graph $G=(V,E)$, first introduced in [4] and defined as the counting function of colorings $c:V\to[x]:=\left\{1,2,\dots,x\right\}$ that satisfy for any edge $vw\in E$

[TABLE]

The usual univariate chromatic polynomial of $G$ can be recovered as the special evaluation $\chi_{G}(x,x)$. Dohmen, Pönitz, and Tittmann provided basic properties of $\chi_{G}(x,y)$ in [4], including polynomiality and special evaluations yielding the matching and independence polynomials of $G$. Subsequent results include a deletion–contraction formula and applications to Fibonacci-sequence identities [5], common generalizations of $\chi_{G}(x,y)$ and the Tutte polynomial [1], and closed formulas for paths and cycles [3].

For a finite poset $(P,\preceq)$, Stanley [6] (see also [8, Chapter 3]) famously introduced the “chromatic-like” order polynomial $\Omega_{P}(x)$ counting all order preserving maps $\varphi:P\to[x]$, that is,

[TABLE]

Here we think of $[x]=\{1,2,\dots,x\}$ as a chain with $x$ elements, and so $\leq$ denotes the usual order in $\mathbb{R}$. The connection to chromatic polynomials is best exhibited through a variant of $\Omega_{P}(x)$, namely the number $\Omega_{P}^{\circ}(x)$ of all strictly order preserving maps $\varphi:P\to[x]$:

[TABLE]

When thinking of $P$ as an acyclic directed graph, it is a short step interpreting $\Omega_{P}^{\circ}(x)$ as a directed version of the chromatic polynomial. Along the same lines, one can write the chromatic polynomial of a given graph $G$ as

[TABLE]

Stanley’s two main initial results on order polynomials were

• •

decomposition formulas for $\Omega_{P}(x)$ and $\Omega_{P}^{\circ}(x)$ in terms of certain permutation statistics for linear extensions of $P$, from which polynomiality of $\Omega_{P}(x)$ and $\Omega_{P}^{\circ}(x)$ also follows;

• •

the combinatorial reciprocity theorem $(-1)^{|P|}\,\Omega_{P}(-x)\ =\ \Omega_{P}^{\circ}(x)\,.$

The latter, combined with (1), gives in turn rise to

• •

Stanley’s reciprocity theorem for chromatic polynomials: $(-1)^{|V|}\,\chi_{G}(-x)$ equals the number of pairs of an $x$-coloring and a compatible acyclic orientation [7].

Our goal is to extend these three results to the bivariate setting.111 A reciprocity theorem for the bivariate chromatic polynomial was previously given in [2]; unfortunately, its statement and proof are wrong.

Along the way, we introduce bivariate versions of the order polynomials which, we believe, are interesting in their own right.

We will call a poset $(P,\preceq)$ bicolored if we think of $P$ as the disjoint union of $C$ and $S$, which we call the celeste and silver elements of $P$. We call $\varphi:P\to[x]$ an order preserving $(x,y)$-map if

[TABLE]

Moreover, $\varphi:P\to[x]$ is a strictly order preserving $(x,y)$-map if

[TABLE]

The functions $\Omega_{P,\,C}(x,y)$ and $\Omega^{\circ}_{P,\,C}(x,y)$ count the number of order preserving $(x,y)$-maps and strictly order preserving $(x,y)$-maps, respectively. We note that these are natural extensions of the single-variable order polynomials, as $\Omega_{P}(x)=\Omega_{P,\,C}(x,1)$ and $\Omega_{P}^{\circ}(x)=\Omega_{P,\,C}^{\circ}(x,0)$. It is a priori not clear why $\Omega_{P,\,C}(x,y)$ and $\Omega^{\circ}_{P,\,C}(x,y)$ are polynomials in two variables, but the following simple example gives an indication.

Example 1**.**

Consider the bicolored chain $P=\{a_{1},a_{2}\}$ where $a_{1}\prec a_{2}$ and $a_{2}$ is celeste. We would like to find all possible strictly order preserving $(x,y)$-maps $\varphi:P\rightarrow[x]$ such that $\varphi(a_{2})>y$. The first case

[TABLE]

gives $y$ choices for $\varphi(a_{1})$ and $x-y$ choices for $\varphi(a_{2})$, whereas the second case

[TABLE]

yields $\binom{x-y}{2}$ choices. Therefore $\Omega^{\circ}_{P,\,C}(x,y)=\frac{1}{2}(x^{2}-x-y^{2}-y)$.

We now give an overview of the main results in this paper. After setting up machinery for bicolored chains in Section 2, we will prove decomposition formulas for $\Omega_{P,\,C}(x,y)$ and $\Omega^{\circ}_{P,\,C}(x,y)$ in terms of linear extensions of $P$ in Section 3 (see Theorem 7 below), as well as:

Theorem 1**.**

The functions $\Omega_{P,\,C}(x,y)$ and $\Omega^{\circ}_{P,\,C}(x,y)$ are polynomials in $x$ and $y$. They satisfy the reciprocity theorem

[TABLE]

To state our reciprocity theorem for bivariate chromatic polynomials, which we will prove in Section 4, we recall that a flat of a graph $G$ is a graph $H$ that can be constructed from $G$ by a series of contractions. We denote by $V(H)$ the vertex set of $H$, and by $C(H)$ the set of vertices of $H$ that resulted from contractions of $G$.

Theorem 2**.**

For any graph $G=(V,E)$,

[TABLE]

where $m_{H}(\sigma,c)$ is the number of pairs $(\sigma,c)$ consisting of an acyclic orientation $\sigma$ of $H$ and a compatible coloring $c:V(H)\rightarrow[x]$ such that $(v)>y$ if $v\in C(H)$.

Here an orientation and a coloring $c$ are compatible if $c(v)\leq c(w)$ for anyx edge oriented from $v$ to $w$.

2. Bicolored Chains

Lemma 3**.**

If $P=\{a_{1}\prec\cdots\prec a_{k}\prec a_{k+1}\prec\cdots\prec a_{n}\}$ is a bicolored chain of length $n$ where $a_{k+1}$ is the minimal celeste element, then

[TABLE]

Proof.

We start with the count of strictly order preserving maps $\varphi:P\rightarrow[x]$ with $\varphi(a_{k+1})>y$. This gives us a set of inequalities for $\varphi$ with $k+1$ cases: for

[TABLE]

there are $\binom{y}{k}\binom{x-y}{n-k}$ possible maps $\varphi$. The second case

[TABLE]

yields $\binom{y}{k-1}\binom{x-y}{n-k+1}$ possible maps $\varphi$. We repeat this process up to the last case

[TABLE]

for which there are $\binom{y}{0}\binom{x-y}{n}$ possible maps $\varphi$.

The count of the weakly order preserving maps $\varphi:P\rightarrow[x]$ with $\varphi(a_{k+1})\geq y$ follows in a similar fashion. ∎

We remark that this lemma proves polynomiality of $\Omega^{\circ}_{P,\,C}(x,y)$ and $\Omega_{P,\,C}(x,y)$, and we obtain the following precursor of Theorem 1 for chains.

Corollary 4**.**

If $P=\{a_{1}\prec\cdots\prec a_{k}\prec a_{k+1}\prec\cdots\prec a_{n}\}$ is a bicolored chain of length $n$ where $a_{k+1}$ is the minimal celeste element, then

[TABLE]

Proof.

By Lemma 3 and the elementary identity $(-1)^{d}\binom{-n}{d}=\binom{n+d-1}{d}$,

[TABLE]

Next we need a refinement of the above results for certain chains that arise from a given general poset on $n$ elements. A linear extension of the bicolored poset $P$ is a chain $L$ whose underlying set (including the celeste/silver partition) equals that of $P$ such that

[TABLE]

Similar to the situation in classical poset theory, linear extensions are best recorded as permutations of a fixed labeling $\omega:P\rightarrow[n]$. Namely, to a given linear extension $L=\{a_{1}\prec a_{2}\prec\dots\prec a_{n}\}$ of $P$, we associate the word $\omega_{L}:=\omega(a_{1})\omega(a_{2})\cdots\omega(a_{n})$. We collect the ascents and descents of the word $\omega_{L}$ in the sets

[TABLE]

respectively. The cardinality of these sets are denoted by $\operatorname{asc}(\omega_{L}):=|\operatorname{Asc}(\omega_{L})|$ and $\operatorname{des}(\omega_{L}):=|\operatorname{Des}(\omega_{L})|$, respectively.

An order preserving $(x,y)$-map $\varphi$ of $L$ is of type $\omega_{L}$ if for $i\in\operatorname{Asc}(\omega_{L})$ we have $\varphi(a_{i})\leq\varphi(a_{i+1})$, and for $i\in\operatorname{Des}(\omega_{L})$ we have $\varphi(a_{i})<\varphi(a_{i+1})$. Let $\Omega_{\omega_{L}}(x,y)$ denote the number of type-$\omega_{L}$ order preserving $(x,y)$-maps; note that this implies that $\varphi(c)\geq y$ for the minimal celeste element $c\in L$. The definition of $\Omega^{\circ}_{\omega_{L}}(x,y)$ is identical except that now $\varphi(c)>y$.

Example 2**.**

Consider the poset $P$ and labeling $\omega$ in Figure 1. Its three linear extensions $L_{1}$, $L_{2}$, and $L_{3}$ come with the associated words $\omega_{L_{1}}=12345$, $\omega_{L_{2}}=12435$, and $\omega_{L_{3}}=14235$, respectively.

As an example, we will compute the number of type-$\omega_{L_{3}}$ order preserving $(x,y)$-maps $\varphi$. The word $\omega_{L_{3}}=14235$ has a descent at $i=2$. Thus, a type-$14235$ order preserving $(x,y)$-map satisfies the set of inequalities

[TABLE]

We remove equalities by defining a bijection $\varphi\mapsto\bar{\varphi}$ via

[TABLE]

Now (2) becomes

[TABLE]

The additional inequality $\varphi(c)>y$ gives rise to $\bar{\varphi}(c)>y+2$. Lemma 3 now yields

[TABLE]

This example contains essentially all ingredients needed for the proof of our next lemma.

Lemma 5**.**

Let $P$ be a bicolored poset with a fixed labeling $\omega:P\rightarrow[n]$. Given a linear extension $L=\{a_{1}\prec a_{2}\prec\dots\prec a_{n}\}$ of $P$ with associated word $\omega_{L}=\omega({a_{1}})\cdots\omega(a_{n})$, let $a_{k+1}$ be the minimal celeste element in $L$ and denote $\widetilde{\omega}_{L}=\omega({a_{1}})\cdots\omega(a_{k+1})$. Then

[TABLE]

and

[TABLE]

Proof.

We want to count the number of order preserving $(x,y)$-maps $\varphi:L\rightarrow[x]$ of type $\omega_{L}$. The map $\varphi$ satisfies the set of inequalities $\varphi(a_{i})\leq\varphi(a_{i+1})$ if $i$ is an ascent of $\omega_{L}$, and $\varphi(a_{i})<\varphi(a_{i+1})$ if $i$ is a descent of $\omega_{L}$. We eliminate weak inequalities by creating a bijection $\varphi\rightarrow\bar{\varphi}$ defined by $\bar{\varphi}(a_{1})=\varphi(a_{1})$ and

[TABLE]

If $i\in\operatorname{Des}(\omega_{L})$, then $\operatorname{asc}\left(\omega(a_{1})\cdots\omega(a_{i+1})\right)-\operatorname{asc}\left(\omega(a_{1})\cdots\omega(a_{i})\right)=0$, and so

[TABLE]

If $i\in\operatorname{Asc}(\omega_{L})$, then $\operatorname{asc}\left(\omega(a_{1})\cdots\omega(a_{i+1})\right)-\operatorname{asc}\left(\omega(a_{1})\cdots\omega(a_{i})\right)=1$, whence

[TABLE]

In addition, $\varphi(a_{k+1})>y$ implies $\bar{\varphi}(a_{k+1})>y+\operatorname{asc}(\widetilde{\omega}_{L})$, and with

[TABLE]

the formula for $\Omega_{\omega_{L}}^{\circ}(x,y)$ now follows from Lemma 3.

The formula for $\Omega_{\omega_{L}}(x,y)$ follows in an analogous fashion, replacing all ascents in our argumentation by descents: now our bijection is defined via $\bar{\varphi}(a_{1})=\varphi(a_{1})$ and

[TABLE]

giving rise to $\bar{\varphi}(a_{k+1})\geq y+\operatorname{des}(\widetilde{\omega}_{L})$ and

[TABLE]

The inverse of $\omega=\omega_{1}\cdots\omega_{n}$ is the word $\overline{\omega}$ defined by $\overline{\omega}_{j}:=\omega_{n+1-j}$. This switches all ascents with descents; in particular, $\operatorname{asc}(\omega_{L})=\operatorname{des}(\overline{\omega}_{L})$ and $\operatorname{des}(\omega_{L})=\operatorname{asc}(\overline{\omega}_{L})$.

Corollary 6**.**

Let $P$ be a bicolored poset with a fixed labeling $\omega:P\rightarrow[n]$, and let $L=\{a_{1}\prec a_{2}\prec\dots\prec a_{n}\}$ be a linear extension of $P$, with associated word $\omega_{L}=\omega({a_{1}})\cdots\omega(a_{n})$. Then

[TABLE]

Proof.

Let $a_{k+1}$ be the minimal celeste element in $L$ and let $\widetilde{\omega}_{L}=\omega(a_{1})\cdots\omega(a_{k+1})$. By Lemma 5 and the fact that $\operatorname{asc}(\widetilde{\omega}_{L})=\operatorname{des}(\overline{\widetilde{\omega}}_{L})$,

[TABLE]

3. Bicolored Posets

To apply Lemma 5 to general posets, we recall the notions of a natural labeling $\omega:P\rightarrow[n]$ of a poset $P$ with $n$ elements, that is,

[TABLE]

and a reverse natural labeling:

[TABLE]

Theorem 7**.**

Let $P$ be a bicolored poset with a reverse natural labeling $\omega$ and a natural labeling $\tau$. Then

[TABLE]

where both sums are over all possible linear extensions $L$ of $P$.

Proof.

We will show that there is a bijection between the set of strictly order preserving $(x,y)$-maps $\varphi$ and the set of all pairs $(L,\hat{\varphi})$ where $L$ is a linear extension and $\hat{\varphi}$ is a type-$\omega_{L}$ map on the bicolored poset $P$.

Suppose $\varphi$ is a strictly order preserving $(x,y)$-map. We will build a linear extension $L$ for which $\varphi$ is of type $\omega_{L}$.

• •

If $\varphi(a_{i})<\varphi(a_{j})$, then let $a_{i}\prec a_{j}$ in $L$. Note that $\omega(a_{i})>\omega(a_{j})$ because $\omega$ is a natural reverse labeling.

• •

If $\varphi(a_{i})=\varphi(a_{j})$ with $\omega(a_{i})<\omega(a_{j})$, then let $a_{i}\prec a_{j}$ in $L$.

• •

If $\varphi(a_{i})=\varphi(a_{j})$ with $\omega(a_{i})>\omega(a_{j})$, then let $a_{i}\succ a_{j}$ in $L$.

This gives us the linear extension $L$ with associated word $\omega_{L}$, and by construction, $\varphi$ is of type $\omega_{L}$.

Conversely, let $L$ be a linear extension of $P$ and let $\hat{\varphi}$ be a type-$\omega_{L}$ map. Type-$\omega_{L}$ maps agree with strictly order preserving maps between comparable elements in a poset by definition. The map $\hat{\varphi}$ may give us weak or strict inequalities between incomparable elements in $P$. Strictly order preserving maps place no restrictions on what happens between incomparable elements in the poset. Thus, $\hat{\varphi}$ is also a strictly order preserving map on $P$.

The proof of the second part is similar to that of the first part with one major difference: now we have $\tau(a_{i})<\tau(a_{j})$ when $a_{i}\prec a_{j}$ in the linear extension $L$ because $\tau$ is a natural labeling. ∎

Proof of Theorem 1.

Polynomiality follows from Lemma 5 and Theorem 7.

To prove the reciprocity statement, fix a natural reverse labeling $\omega$ of $P$; note that $\overline{\omega}$ is a natural labeling. By Corollary 6 and Theorem 7,

[TABLE]

4. Bivariate Chromatic Polynomials

In preparation for our proof of Theorem 2, we show the following decomposition result.

Lemma 8**.**

[TABLE]

Proof.

Let $c:V\to[x]:=\left\{1,2,\dots,x\right\}$ be a coloring of $G$ that satisfies

[TABLE]

Let $H$ be the flat of $G$ obtained by contracting all edges whose endpoints have the same color; note that the colors of $C(H)$ will be $>y$. We orient the edges of $H$ by their color gradient: $v\to u$ if $c(v)<c(u)$. The resulting orientation will be acyclic, and the resulting colorings of $H$ will precisely be counted by $\Omega^{\circ}_{\sigma,\,C(H)}(x,y)$.

This argumentation can be reversed: given a flat $H$ of $G$ and an acyclic orientation $\sigma$ of $H$, a strictly order preserving $(x,y)$-map counted by $\Omega^{\circ}_{\sigma,\,C(H)}(x,y)$ can be extended to a coloring of $G$ where vertices that result in contractions for $H$ get colors $>y$. ∎

Example 3**.**

Our proof of Lemma 8 is illustrated for the case $G=K_{3}$ in Figures 2 and 3.

For $H=G$ we obtain $\Omega^{\circ}_{\sigma,\,\varnothing}(x,y)=\binom{x}{3}$ for any acyclic orientation $\sigma$, and summing over all six such orientations gives the univariate chromatic polynomial of $K_{3}$.

If $H$ is the result of one contraction, we need to consider two orientations $\sigma_{1}$ and $\sigma_{2}$ with order polynomials

[TABLE]

Finally, if $H$ results from contracting two edges, $\Omega^{\circ}_{\sigma,\,v_{123}}(x,y)=x-y$. Thus

[TABLE]

Proof of Theorem 2.

By Theorem 1 and Lemma 8,

[TABLE]

Here $\Omega_{\sigma,\,C(H)}(x,y+1)$ counts the number of order preserving maps $\varphi:\sigma\rightarrow[x]$ subject to two conditions: First, if $c\in C(H)$ then $\varphi(c)\geq y+1$; and second, $\varphi$ is compatible with $\sigma$. ∎