Chromatic symmetric functions from the modular law

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Abstract

In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced in [19]. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic quasisymmetric function in the elementary basis are positive unimodal polynomials and characterize them as certain q-hit numbers (up to a factor). Finally, we discuss the logarithmic concavity of the coefficients of the chromatic quasisymmetric function.

Introduction

The chromatic polynomial can be characterized as the unique functionχ:GraphsQ[x] that has the following three properties.1

  • (A)

    It satisfies the deletion-contraction recurrence, χG=χGeχG/e for every edge eE(G).

  • (B)

    It is multiplicative, χG1G2=χG1χG2.

  • (C)

    It has values at complete graphs given by χKn(x)=x(x1)(xn+1).

The chromatic polynomial of a graph admits a symmetric function generalization introduced by Stanley in [31]. Given a graph G it is defined ascsf(G):=κxκ where the sum runs through all proper colorings of the vertices κ:V(G)N and xκ:=vV(G)xκ(v). A coloring κ is proper if κ(v)κ(v) whenever v and v are adjacent. If Λ is the algebra of symmetric functions, it turns out that csf is a function from Graphs to Λ. This function is multiplicative and its values at complete graphs are given by csf(Kn)=n!en (where en is the elementary symmetric function of degree n). However, it does not satisfy the deletion-contraction recurrence, one simple reason being that the chromatic symmetric function is homogeneous of degree equal to the number of vertices of G.

In this paper we will restrict ourselves to indifference graphs, i.e., graphs whose set of vertices can be identified with [n]:={1,2,,n} and such that if {i,j} is an edge with i<j, then {i,k} and {k,j} are also edges for every k such that i<k<j. This class of graphs, restrictive as it might look, ends up having deep relations with geometry and representation theory, see, for example, [8], [20] and [1].

Indifference graphs can be naturally associated with Hessenberg functions and Dyck paths (see Fig. 1A). A Hessenberg function is a non-decreasing function h:[n][n] such that h(i)i for every i[n]. The graph associated to h is the graph with vertex set [n] and set of edges E={{i,j};i<jh(i)}. All indifference graphs arise from Hessenberg functions. To each Hessenberg function there is an associated Dyck path, which is the unique path with h(i) north steps before the i-th east step. We usually denote a Hessenberg function by the n-tuple of its values h=(h(1),h(2),,h(n)) or by the word in n (north step) and e (east step) corresponding to its associated Dyck path. We denote by D the set of Dyck paths, which will be identified with the set of Hessenberg functions and with the set of indifference graphs. In the rest of the introduction by graph we will always mean an indifference graph.

When G is the graph associated to h, we can recover the chromatic polynomial of G by a differential operator Δh in the Weyl algebra Q[x,]. The operator Δh is obtained from h by replacing each east step with ∂ and each north step with x. For example, if h=nnenee, then Δh=x2x2. It is not hard to check that the chromatic polynomial of G satisfies the following equalityΔhxn=χG(n)xn.

Moreover, with this interpretation, the deletion-contraction recurrence (when applied to edges of G that correspond to corners in h) is, essentially, the well known formula [,x]=1, which givesx=x1. See Fig. 1B.

As pointed out before, the fact that Formula (1a) is not homogeneous is one of the reasons for the deletion-contraction recurrence not holding for the chromatic symmetric function. On the other hand, it is not hard to find homogeneous relations for ∂ and x. For example, one can simply consider [[,x],x]=0 and [[,x],]=0. Explicitly this gives2xx=x2+x22x=2x+x2. In particular, if G0, G1 and G2 are graphs associated to Dyck paths h0, h1 and h2 such that h0 and h2 are obtained from h1 by replacing a subpath nen with enn and nne, respectively, then (see Fig. 1C)

2χG1=χG0+χG2. Similarly, the same holds if h0 and h2 are obtained from h1 by replacing a subpath ene with een and nee, respectively.

One can actually replace Property (A) (the deletion-contraction recurrence) with the recurrence in Equation (1c), and these properties will still characterize the chromatic polynomial for indifference graphs. In other words, the restriction of χ to the set of indifference graphs is the unique function that has the following three properties.

  • (A')

    Whenever G0, G1, and G2 are graphs associated to Dyck paths h0, h1, and h2 such that h0 and h2 are obtained from h1 by replacing a subpath nen with enn and nne, respectively, or by replacing a subpath ene with een and nee, respectively, then 2χG1=χG0+χG2.

  • (B)

    It is multiplicative, χG1G2=χG1χG2.

  • (C)

    It has values at complete graphs given by χKn(x)=x(x1)(xn+1).

One possible way to see this is to find more general relations between ∂ and x starting from Equation (1b). One example is the following(b+1)xlxb=ll1xb+1+(b+1l)lxb+1. This equation, translated to graphs, means that when G0, G1 and G2 are the graphs associated to Dyck paths h0, h1, and h2 (see Fig. 2C) such that h0 and h2 are obtained from h1 by replacing a subpath nelnb with elnb+1 and el1nb+1e, respectively, then we have(b+1)χG1=lχG2+(b+1l)χG0.

Given a connected graph G1, which is not complete, we have that its associated Dyck path h1 must end with nelnbec for some positive integers l,b,c. If we define h0 and h2 by replacing the subpath nelnb of h1 with elnb+1 and el1nb+1e, respectively, we have that both h0 and h2 are Dyck paths (because G1 is connected) and we can apply Equation (1e). This process eventually ends, since b will increase at each step. Of course, the chromatic polynomial of indifference graphs can be easily computed directly, but the above recurrence is useful, for instance, if one wants to write the chromatic polynomial in certain bases as in [7].

The idea is to repeat this process for the chromatic symmetric function. Actually, we will work with the chromatic quasisymmetric function introduced by Shareshian and Wachs in [29]. For a graph G with set of vertices [n], the chromatic quasisymmetric function csfq(G) is defined ascsfq(G):=κqascG(k)xκ, where the sum runs through all proper colorings of G andascG(κ):=|{(i,j);i<j,κ(i)<κ(j);{i,j}E(G)}| is the number of ascents of the coloring κ. When G is an indifference graph, we have that csfq(G) is actually symmetric, so we will think of csfq as function csfq:DΛq, where Λq is the algebra of symmetric functions with coefficients in Q(q).

With a little experimentation one can see that Equation (1c) does not hold in general for the chromatic symmetric function of indifference graphs. However, it still holds if we add some extra assumptions on h0, h1 and h2, which are summarized in Definition 2.1. The purpose of this article is to determine when Equation (1e) lifts to the chromatic symmetric function (see Proposition 2.4). Moreover, we prove that there are enough of these liftings to fully characterize Stanley's chromatic symmetric function on indifference graphs as stated in the following theorem. As usual, we define [n]q:=qn1q1 and n!q:=j=1n[j]q.

Theorem 1.1

The function csfq:DΛq is the unique function that has the following three properties.

  • (A)

    It satisfies the modular law, as in Definition 2.1.

  • (B)

    It is multiplicative, csfq(G1G2)=csfq(G1)csfq(G2).

  • (C)

    It has values at complete graphs given by csfq(Kn)=n!qen.

Actually we prove a more general result. For an indifference graph G with vertex set [n], we denote by Gt its transposed graph, that is, we relabel the vertices of G via in+1i.

Theorem 1.2

Let A be a Q(q)-algebra and let f:DA be a function that satisfies the modular law, as in Definition 2.1. Then f is determined by its values f(Kn1Kn2Knm) at the disjoint ordered union of complete graphs and these values are independent of the order in which the union is taken. Moreover, we have that f(Gt)=f(G) for every indifference graph G.

Our proof of Theorem 1.2 is constructive. We find an algorithm (Algorithm 2.8) based exclusively on the modular law. This algorithm was implemented in SAGE and is available upon request.

Every function that satisfies the modular law is intimately related with the chromatic symmetric function, as seen in Corollary 3.2. It would be interesting to find functions with combinatorial interpretations that satisfy the modular law. In [2] the authors define one such function enumerating increasing spanning forests and use it to sharpen the description of the e-coefficients of unicellular LLT polynomials conjectured in [4] and [16].

When G is an indifference graph whose complement is bipartite, we observe in Remark 2.12 that the algorithm reduces to a planar network. In this situation, we show that csfq(G) can be computed in terms of q-hit numbers. This is a q-analogue of the Stanley-Stembridge combinatorial formula [32, Theorem 4.3]. We recall that the partition associated to h is given by λ=(nh(1),nh(2),,nh(n)).

Theorem 1.3

Let h be an Hessenberg function whose associated indifference graph has bipartite complement and let λ be the partition associated to h. If m:=min{λ1,(λ)}, thencsfq(h)=m!qRm,nm(λ)enm,m+j<mqjj!q[m2j]qRj,nj1(λ)enj,j. where Rj,k(λ) are the Garsia-Remmel q-deformation of hit numbers, that is, it enumerates weighted rook placements on k×k board with exactly j rooks on the Young diagram of λ. Moreover, we have that csfq(h) is e-unimodal.

In the last section we discuss the logarithmic concavity of the coefficients of the chromatic quasisymmetric function. In the breakthrough work [25] it is proved that the chromatic polynomial of a graph is log-concave. This result was later generalized to matroids in [3]. We supply some evidence supporting the logarithmic concavity of the e-coefficients of csfq(h) for hD.

We point out that several analogues of deletion-contraction exist for the chromatic symmetric function (or some closely related symmetric functions). A non-commutative chromatic symmetric function is defined in [17] which satisfies a deletion-contraction recurrence. In [19] and [27], a modular law for the chromatic symmetric function is introduced for any graph. When restricted to indifference graphs it is the analogue of Equation (1c). In [26], this relation is found for the closely related unicellular LLT polynomial. A chromatic symmetric function for weighted graphs is defined in [12] which satisfies a deletion-contraction recurrence when one considers contractions of weighted graphs. Other linear relations in various settings can be found in [27], [24], [13], and [6].

It is also worth mentioning that, for unicellular LLT polynomials, the analogy with differential operators was made precise in [9]. They defined operators d and d+ which play the roles of x and ∂ in the discussion above, and proved that the unicellular LLT polynomial associated to a Dyck path h can be computed as dh(1) where dh is the operator obtained from h by replacing each east step with d+ and each north step with d.

In the recent paper [6] it is demonstrated how to obtain the chromatic symmetric function from a similar set of relations (see [6, Corollary 6.16]). Actually, one of the relations used in [6] is contained in the modular law used here. The authors also consider a bounce relation on Schröeder paths that implies the modular law and other relations.

Acknowledgments

We thank the anonymous referees for the careful reading of the paper and constructive suggestions.

Section snippets

The algorithm

Our main goal in this section is to prove Theorem 1.2. We first need some notation. We denote the set of Dyck paths by D and by Dn the set of Dyck paths of size n. We will also think of D as the set of Hessenberg functions via the identification between Dyck paths and Hessenberg functions. There is a (non-commutative) product on the set D given by concatenation of Dyck paths, while on Hessenberg functions the product of h1:[n1][n1] with h2:[n2][n2] is the function h:[n1+n2][n1+n2] given by h(

The chromatic quasisymmetric function

We begin by recalling that the chromatic quasisymmetric function does indeed satisfy the modular law. This is already well known in the literature (see for instance [19, Proposition 3.1], [26, Theorem 3.4], [24, Theorem 3.1] and [4, Corollary 20 and Proposition 23]). In particular, if we writecsfq(h)=λncsfq,λ(h)eλ we have that the functions csfq,λ:DQ(q) also satisfy the modular law. The following result proves that every other function f:DQ(q) is actually a Q(q)-linear combination of csfq,λ.

Rook placements and q-hit numbers

In this section, we prove a q-analogue of [32, Theorem 4.3]. We mention that rook placements also appear in relation with terms in the e-expansion of csfq, see [5]. First we must recall the definition of the q-analogue of hit numbers introduced in [15].

Let Em be the m×m board and let λ be a partition such that its Young diagram fits in Em, that is λ1,(λ)m. To keep the notation consistent with the last sections, we will number the rows of Em from bottom to top. This means that the Young

Logarithmic concavity

In this section we discuss the logarithmic concavity of the coefficients of csfq. We need a few definitions first. We say that a polynomial P(q)=ajqj is log-concave with no internal zeros if it is unimodal and aj2aj1aj+1 for every j. For simplicity, in what follows we will write log-concave instead of log-concave with no internal zeros. It is a well know fact that the product of two log-concave polynomials still is a log-concave polynomial (see [30]). On the other hand, the sum of two

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