CCHROMATIC QUASISYMMETRIC FUNCTIONS OF DIRECTEDGRAPHS
BRITTNEY ELLZEY Abstract.
Chromatic quasisymmetric functions of labeled graphs were defined byShareshian and Wachs as a refinement of Stanley’s chromatic symmetric functions.In this extended abstract, we consider an extension of their definition from labeledgraphs to directed graphs, suggested by Richard Stanley. We obtain an F -basisexpansion of the chromatic quasisymmetric functions of all digraphs and a p -basisexpansion for all symmetric chromatic quasisymmetric functions of digraphs, extend-ing work of Shareshian-Wachs and Athanasiadis. We show that the chromatic qua-sisymmetric functions of proper circular arc digraphs are symmetric functions, whichgeneralizes a result of Shareshian and Wachs on natural unit interval graphs. The di-rected cycle on n vertices is contained in the class of proper circular arc digraphs, andwe give a generating function for the e -basis expansion of the chromatic quasisym-metric function of the directed cycle, refining a result of Stanley for the undirectedcycle. We present a generalization of the Shareshian-Wachs refinement of the Stanley-Stembridge e -positivity conjecture. Introduction
Let G = ( V, E ) be a (simple) graph. A proper coloring, κ : V → P , of G is anassignment of positive integers, which we can think of as colors, to the vertices ofG such that adjacent vertices have different colors; in other words, if { i, j } ∈ E, then κ ( i ) (cid:54) = κ ( j ) . The chromatic polynomial of G, denoted χ G ( k ) , gives the number of propercolorings of G using k colors. Stanley [Sta95] defined a symmetric function refinementof the chromatic polynomial, called the chromatic symmetric function of a graph. If welet the vertex set of G be V = { v , v , · · · v n } , then the chromatic symmetric functionof G is defined as X G ( x ) = (cid:88) κ x κ ( v ) x κ ( v ) · · · x κ ( v n ) where the sum ranges over all proper colorings, κ , of G and κ ( v i ) denotes the colorof v i . The chromatic symmetric function of a graph refines the chromatic polynomial,because if we replace x , x , · · · , x k with 1’s and all other variables with 0’s, then X G (1 , , · · · , , , , · · · ) = χ G ( k ) . We can easily see that for any graph G, X G ( x ) ∈ Λ Q , where Λ Q is the Q -algebra ofsymmetric functions in the variables x , x , · · · with coefficients in Q . For any basis, { b λ | λ (cid:96) n } , of Λ Q , we say that a symmetric function, f ∈ Λ Q is b -positive if the Date : November 14, 2016; revised April 1, 2017. Supported in part by NSF Grant DMS 1202337. a r X i v : . [ m a t h . C O ] A p r BRITTNEY ELLZEY expansion of the function in terms of the b λ -basis has positive coefficients. The sym-metric function bases we focus on in this paper are the power sum symmetric functionbasis, p λ , and the elementary symmetric function basis, e λ . We assume the reader isfamiliar with the basic theory of symmetric and quasisymmetric functions, which canbe found in [Sta99].Stanley [Sta95] proves that ωX G ( x ) is p -positive for all graphs, G , where ω is theusual involution on Λ Q . A long-standing conjecture on chromatic symmetric functionsinvolves their e -positivity. Recall that a poset is ( a + b )-free if it has no induced posetthat consists of a chain of length a and a chain of length b. The incomparability graph ofa poset P, denoted inc ( P ) , is the graph whose vertices are the elements of P and whoseedges correspond to pairs of incomparable elements of P. The following conjecture is ageneralization of a particular case of a conjecture of Stembridge on immanants [Ste92]. Conjecture 1 (Stanley-Stembridge [Sta95]) . Let P be a (3 + 1) -free poset. Then X inc ( P ) ( x ) is e -positive. For subsequent work on chromatic symmetric functions, see the references in [SW16].Shareshian and Wachs [SW12, SW16] defined a quasisymmetric refinement of Stan-ley’s chromatic symmetric function called the chromatic quasisymmetric function of a(labeled) graph. Let G = ([ n ] , E ) and let κ : [ n ] → P be a proper coloring of G. Definethe ascent number of κ asasc( κ ) = |{{ i, j } ∈ E | i < j, κ ( i ) < κ ( j ) }| . The chromatic quasisymmetric function of G is defined as X G ( x , t ) = (cid:88) κ t asc( κ ) x κ (1) x κ (2) · · · x κ ( n ) where κ ranges over all proper colorings of G. Notice that setting t = 1 reduces thisdefinition to Stanley’s original chromatic symmetric function.In the Shareshian-Wachs chromatic quasisymmetric function of a graph, it is nothard to see that the coefficient of t j for each j ∈ N is a quasisymmetric function;however, the coefficients do not have to be symmetric. If G is the path 1 − − , then X G ( x , t ) has symmetric coefficients, i.e. X G ( x , t ) ∈ Λ Q [ t ], but if G is the path 1 − − ,X G ( x , t ) does not have symmetric coefficients (see [SW16, Example 3.2]). In general, X G ( x , t ) ∈ QSym Q [ t ] , where QSym Q [ t ] is the ring of polynomials in t with coefficientsin the ring of quasisymmetric functions in the variables x , x , · · · with coefficients in Q . Shareshian and Wachs show that if G is natural unit interval graph (that is, a unitinterval graph with a certain natural labeling), then X G ( x , t ) ∈ Λ Q [ t ] . For G a naturalunit interval graph, they show that the coefficient of each power of t in X G ( x , t ) is Schur-positive, and they conjecture that these coefficients are e -positive and e -unimodal. Infact, Guay-Paquet [GPa] shows that if the Stanley-Stembridge conjecture holds forunit interval graphs, then the conjecture holds in general. Hence the Shareshian-Wachs e -positivity conjecture implies the Stanley-Stembridge conjecture. Shareshian HROMATIC QUASISYMMETRIC FUNCTIONS OF DIRECTED GRAPHS 3 and Wachs present a formula for the e -basis expansion of X P n ( x , t ) , where P n is thepath on n vertices with a natural labeling, showing that X P n ( x , t ) is e -positive.Shareshian and Wachs also conjectured a formula for the p -basis expansion of ωX G ( x , t ) , where G is a natural unit interval order, which would imply that ωX G ( x , t ) is p -positive.Athanasiadis [Ath15] later proved this formula.There is an important connection between chromatic quasisymmetric functions ofnatural unit interval graphs and Hessenberg varieties, which was conjectured by Shareshianand Wachs and was proven by Brosnan and Chow [BC] and later by Guay-Paquet[GPb]. Clearman, Hyatt, Shelton, and Skandera [CHSS16] found an algebraic inter-pretation of chromatic quasisymmetric functions of natural unit interval graphs interms of characters of type A Hecke algebras evaluated at Kazhdan-Lusztig basis ele-ments. Recently, Haglund and Wilson [HW] discovered a connection between chromaticquasisymmetric functions and Macdonald polynomials.In this paper , we extend the work of Shareshian and Wachs by considering chromaticquasisymmetric functions of (simple) directed graphs . For notational convenience, wedistinguish an undirected graph, G, from a digraph, −→ G , with an arrow.
Definition 2.
Let −→ G = ( V, E ) be a digraph. Given a proper coloring, κ : V → P of −→ G , we define the ascent number of κ asasc( κ ) = |{ ( v i , v j ) ∈ E | κ ( v i ) < κ ( v j ) }| , where ( v i , v j ) is an edge directed from v i to v j . Then the chromatic quasisymmetricfunction of −→ G is defined to be X −→ G ( x , t ) = (cid:88) κ t asc( κ ) x κ ( v ) x κ ( v ) · · · x κ ( v n ) where the sum is over all proper colorings, κ, of −→ G .As with the Shareshian-Wachs chromatic quasisymmetric function, setting t = 1gives Stanley’s chromatic symmetric function. We can easily see that for any digraph, X −→ G ( x , t ) ∈ QSym Q [ t ] . Notice that if we take a labeled graph G = ([ n ] , E ) and make adigraph, −→ G , by orienting each edge from the vertex with the smaller label to the vertexwith the larger label, then X G ( x , t ) = X −→ G ( x , t ) . In other words, this definition of thechromatic quasisymmetric function of a digraph is equivalent to the Shareshian-Wachschromatic quasisymmetric function in the case of an acyclic digraph.In this paper, we present an expansion of ωX −→ G ( x , t ) in Gessel’s fundamental qua-sisymmetric basis with positive coefficients for every digraph, −→ G .
We determine a classof digraphs for which X −→ G ( x , t ) ∈ Λ Q [ t ] , namely proper circular arc digraphs. A sim-ple example of a proper circular arc digraph is −→ C n , the cycle on n vertices orientedcyclically. If we turn natural unit interval graphs into directed unit interval graphs i.e. in the full version of this paper [Ell] See Remark 3.
BRITTNEY ELLZEY by orienting edges from smaller label to larger label, then these directed unit intervalgraphs are contained in the class of proper circular arc digraphs as well and hence oursymmetry result generalizes the result of Shareshian and Wachs.We present a p -positivity result for all digraphs −→ G such that X −→ G ( x , t ) ∈ Λ Q [ t ] , whichdoes not reduce to the formula in the acyclic case conjectured by Shareshian-Wachs[SW16] and proved by Athanasiadis [Ath15]. We give a factorization of the coeffi-cients of z − λ p λ in ωX −→ C n ( x , t ) . We also present a few results on e -positivity, includinga generating function formula for X −→ C n ( x , t ) , which is a t -analog of a result of Stanley[Sta95, Proposition 5.4] and shows its e -positivity. We present a generalization of theShareshian-Wachs e -positivity conjecture for proper circular arc digraphs. We also givea combinatorial interpretation of the coefficients in the elementary symmetric functionexpansion of the chromatic quasisymmetric functions of the cycle, oriented cyclically,and the path, oriented in one direction. Remark . The idea of extending chromatic quasisymmetric functions to directedgraphs was a suggestion made by Richard Stanley to the author after attending atalk on her work on the chromatic quasisymmetric function of the labeled cycle [Ell].Subsequent to our work, Alexandersson and Panova [AP] independently obtainedthe symmetry result of Section 4 and the results of Section 5. However, their proof ofTheorem 12, giving the e -expansion of X −→ C n ( x , t ), is very different from ours.2. Expansion in the Fundamental Quasisymmetric Basis
For incomparability graphs of posets, Shareshian and Wachs give an expansion of ωX G ( x , t ) into Gessel’s fundamental quasisymmetric basis, which shows that these ωX G ( x , t ) are F -positive. We extend this result by presenting an F -basis expansion of ωX −→ G ( x , t ) for all digraphs, which shows that ωX −→ G ( x , t ) is F -positive for all digraphs.In general our formula does not reduce to the formula of Shareshian and Wachs, so thisgives another combinatorial description of the coefficients in the F -basis for incompa-rability graphs of posets.Let −→ G be a digraph and let σ ∈ S n . Defineinv −→ G ( σ ) = |{ ( i, j ) ∈ E ( −→ G ) | σ − ( j ) < σ − ( i ) }| , i.e. the number of ( i, j ) pairs such that j comes before i in σ and there is a directededge from i to j in −→ G .
Now let G = ([ n ] , E ) be an undirected graph and let σ = σ σ · · · σ n ∈ S n . For each x ∈ [ n ] , define rank ( G,σ ) ( x ) as the length of the longest subword σ i σ i · · · σ i k such that i < i < · · · < i k , σ i k = x , and for each 1 ≤ j < k , { σ i j , σ i j +1 } ∈ E . We say σ has a G-descent at i if either rank ( G,σ ) ( σ i ) > rank ( G,σ ) ( σ i +1 ) or rank ( G,σ ) ( σ i ) = rank ( G,σ ) ( σ i +1 )and σ i > σ i +1 . Let DES G ( σ ) be the set of G-descents of σ. For example let G = C , the cycle on 9 vertices labeled cyclically with 1 , , ..., σ = 234658971 ∈ S . For x ∈ [9] , if rank ( G,σ ) ( x ) = 1 , then x = 2 , , . If HROMATIC QUASISYMMETRIC FUNCTIONS OF DIRECTED GRAPHS 5 rank ( G,σ ) ( x ) = 2 , then x = 3 , , . If rank ( G,σ ) ( x ) = 3 , then x = 1 , . If rank ( G,σ ) ( x ) = 4 , then x = 5 . From this we see that DES G ( σ ) = { , , } . Theorem 4.
Let −→ G = ([ n ] , E ) be any directed graph. Then ωX −→ G ( x , t ) = (cid:88) σ ∈ S n F n, DES G ( σ ) ( x ) t inv −→ G ( σ ) where F n,S ( x ) is Gessel’s fundamental quasisymmetric function and G is the underlyingundirected graph of −→ G .
Note that our formula requires that −→ G be labeled with [ n ]; however, the expansionof X −→ G ( x , t ) in the F -basis is the same for any choice of labeling.3. Expansion in the Power Sum Symmetric Function Basis
In [Sta95], Stanley showed that for any graph
G, ωX G ( x ) is p -positive. In [SW16],Shareshian and Wachs conjectured a formula for the p -expansion coefficients that es-tablished that ωX G ( x , t ) is p -positive for any natural unit interval graph, G , and in[Ath15], Athanasiadis proved their conjecture. Here we present a p -positivity result forall digraphs whose chromatic quasisymmetric functions have symmetric coefficients.We give a formula for the coefficients of each p λ in the p -basis expansion of ωX −→ G ( x , t ) . We can assume without loss of generality that the vertex set of −→ G is [ n ] . We want todefine a set of permutations, N G,λ , for every undirected graph G = ([ n ] , E ) and everypartition λ of n.For any word w = w w · · · w k with distinct letters in [ n ] , we say a letter w j with j > G-isolated if for all i < j , there is no edge between w i and w j in G. We define N G,λ as follows. If σ ∈ S n , break σ up into contiguous segments of sizes λ , λ , · · · , λ k (in order) and call these pieces α , α , · · · α k . Then σ ∈ N G,λ means thateach segment α i has no G-isolated letters and does not contain any of the G -descentsof σ . So every permutation is in N G, (1) n , and the permutations of N G, ( n ) are in bijectionwith acyclic orientations of G with a unique sink.In the last section, we determined that for G = C and σ = 234658971 , DES G ( σ ) = { , , } . If we let λ = (3 , , , , , then α = 234 , α = 65 , α = 89 , α = 7 , and α = 1 , and σ ∈ N G,λ ; however if µ = (3 , , , , then 1 is an isolated vertex of α = 71 , so σ / ∈ N G,µ . Theorem 5.
Let −→ G = ([ n ] , E ) be a digraph such that X −→ G ( x , t ) ∈ Λ Q [ t ] . Then ωX −→ G ( x , t ) = (cid:88) λ (cid:96) n z − λ p λ (cid:88) σ ∈ N G,λ t inv −→ G ( σ ) , where G is the underlying undirected graph of −→ G .
Consequently, ωX −→ G ( x , t ) is p-positive. We prove this by first expressing ωX G ( x , t ) in terms of the fundamental quasisym-metric basis, as shown in Theorem 4, and then extending the technique used by BRITTNEY ELLZEY
Athanasiadis [Ath15], which involves the Adin-Roichman [AR15] formula for sym-metric group representations on Schur-positive sets, to prove the acyclic version ofthis result. We point out that our result does not reduce, in an obvious way, to theAthanasiadis-Shareshian-Wachs formula in the acyclic case. It gives a new formula forthis case.In [SW16, Proposition 7.8], Shareshian and Wachs showed that when −→ G is acyclic,the coefficient of each z − λ p λ in ωX −→ G ( x , t ) factors nicely. Though the coefficients of ωX −→ G ( x , t ) do not generally factor in the cyclic case, the coefficient of each z − λ p λ in ωX −→ C n ( x , t ) does have a nice factorization involving the Eulerian polynomials. Theorem 6.
Let −→ C n be the cycle on n vertices directed cyclically and let λ = ( λ , λ , · · · , λ k ) be a partition of n. If k ≥ , then (cid:88) σ ∈ N Cn,λ t inv −→ Cn ( σ ) = ntA k − ( t ) k (cid:89) i =1 [ λ i ] t , where [ n ] t = 1 + t + · · · + t n − and A k ( t ) is the Eulerian polynomial. In the case that λ = ( n ) , we have (cid:88) σ ∈ N Cn, ( n ) t inv −→ Cn ( σ ) = nt [ n − t , Hence the coefficient of n p n in ωX −→ C n ( x , t ) is nt [ n − t and for all other λ (cid:96) n , thecoefficient of z − λ p λ in ωX −→ C n ( x , t ) is ntA k − ( t ) k (cid:89) i =1 [ λ i ] t . Graphs and Symmetry
In this section, we discuss a class of digraphs, −→ G , such that X −→ G ( x , t ) is symmetric.An oriented graph will be called a {−−→ K , −−→ K } -free digraph if it avoids the induceddigraphs −−→ K and −−→ K , shown below.The most well-known class of graphs discussed in this paper are interval graphs .Given a collection of intervals on the real line, we can associate them with a graphby letting each interval correspond to a vertex and each edge correspond to a pair ofoverlapping intervals. Proper interval graphs are interval graphs in which no intervalproperly contains another.
Unit interval graphs are interval graphs in which eachinterval has unit length.
Proposition 7 (Roberts [Rob69], Skrien [Skr82]) . Let G be a graph. Then the fol-lowing statements are equivalent:1. G is a proper interval graph.2. G is a unit interval graph. An oriented graph is a digraph with no bidirected edges.
HROMATIC QUASISYMMETRIC FUNCTIONS OF DIRECTED GRAPHS 7
3. G admits an acyclic orientation that makes it a {−−→ K , −−→ K } -free digraph.The equivalence of (1) and (2) was shown by Roberts [Rob69] and the equivalenceof (1) and (3) was shown by Skrien [Skr82].In [SW16], Shareshian and Wachs prove that chromatic quasisymmetric functionsof labeled graphs have symmetric coefficients for natural unit interval graphs, whichare unit interval graphs with a specific labeling. For example, the path 1 − − − − − − − − −−→ K and the digraph associated with 1 − − −−→ K . For the remainder of the paper, we will use the term unit interval digraphs to refer to natural unit interval graphs viewed as digraphs. It turns out that this isexactly the class of acyclic {−−→ K , −−→ K } -free digraphs.Now let us look at the circular analog of these graph classes. The circular analog ofinterval graphs is the class of circular arc graphs . If we have a collection of arcs on acircle, we can associate a graph to this collection by allowing each arc to correspondto a vertex and each edge to correspond to a pair of overlapping arcs. Proper circulararc graphs are circular arc graphs where no arc properly contains another.
Proposition 8 (Skrien [Skr82]) . Let G be a connected graph. Then the followingstatements are equivalent:1. G is a proper circular arc graph.2. G admits an orientation that makes it a {−−→ K , −−→ K } -free digraph.For the remainder of this paper, we will refer to {−−→ K , −−→ K } -free digraphs as propercircular arc digraphs , because this proposition shows that the underlying undirectedgraph of each of the connected components of a {−−→ K , −−→ K } -free digraph is a propercircular arc graph.The smallest digraphs that do not have symmetric coefficients are −−→ K and −−→ K . Bythe work of Shareshian and Wachs, we know that chromatic quasisymmetric functionsof unit interval digraphs, or acyclic {−−→ K , −−→ K } -free digraphs, have symmetric coeffi-cients. We have the following generalization of this result. The proof closely followsthe proof of the Shareshian-Wachs symmetry result [SW16, Theorem 4.5]. Theorem 9.
Let −→ G be a proper circular arc digraph and let X −→ G ( x , t ) be the chromaticquasisymmetric function associated with the digraph −→ G . Then X −→ G ( x , t ) ∈ Λ Q [ t ] . Note that the converse of this statement is not true. In fact, the cycle with one edgedirected backwards is in Λ Q [ t ]; however, for the rest of this paper, we will focus on thechromatic quasisymmetric functions of proper circular arc digraphs.5. Expansion in the Elementary Symmetric Function Basis
The Shareshian-Wachs conjecture stated in terms of digraphs says that the chromaticquasisymmetric functions of unit interval digraphs are e -positive and e -unimodal, where BRITTNEY ELLZEY we call the palindromic polynomial X G ( x , t ) = (cid:80) mj =0 a j ( x ) t j e-unimodal if a j +1 ( x ) − a j ( x ) is e -positive for 0 ≤ j < m − . We present a generalized version of this conjecture . Conjecture 10.
Let −→ G be a proper circular arc digraph. Then the palindromic poly-nomial X −→ G ( x , t ) is e -positive and e -unimodal. For r, n ∈ N such that 1 ≤ r ≤ n define G n,r = ([ n ] , E ) to be the graph on [ n ] with { i, j } ∈ E if 0 < | i − j | < r. For example, G n, is the graph on [ n ] with no edges,and G n, is the path on [ n ], where consecutive labels are adjacent. It is not difficult tosee that X G n, ( x , t ) = e n . Shareshian and Wachs proved the conjecture for G n,n − and G n,n − [SW16, Corollaries 8.2, 8.3, 8.4] as well as for the path, G n, [SW10, Theorem7.2], and the complete graph, G n,n for all n, and they tested it for all G n,r with n ≤ ≤ r ≤ n, which also implies that this conjecture holds for these G n,r graphsviewed as digraphs rather than labeled graphs.For r, n ∈ N such that 1 ≤ r ≤ (cid:6) n (cid:7) , define the directed circular analog −→ G ∗ n,r =([ n ] , E ) to be the digraph on [ n ] with ( i, j ) ∈ E if 0 < ( j − i ) (mod n ) < r. Forexample, −→ G ∗ n, is the cycle on n vertices with edges directed cyclically. We show in thenext theorem that X −→ G ∗ n, ( x , t ) is e -positive and e -unimodal. We tested the e -positivityand e -unimodality of −→ G ∗ n,r for n ≤ r ≤ (cid:6) n (cid:7) . Note that Conjecture 10 would imply the Shareshian-Wachs conjecture, which in turnwould imply the Stanley-Stembridge conjecture. In addition, Stanley [Sta95] mentionsa class of graphs called circular indifference graphs, which turns out to be equivalentto the class of proper circular arc graphs, as a possible class of graphs with e -positivechromatic symmetric functions. Conjecture 10 refines this conjecture as well.One can generalize [Sta95, Theorem 3.3] of Stanley and [SW16, Theorem 5.3] ofShareshian and Wachs to show that the sums of certain coefficients in the e -basisexpansion are positive. Proposition 11.
Let −→ G be a proper circular arc digraph on n vertices with G as itsunderlying undirected graph. Suppose we have the expansion X −→ G ( x , t ) = (cid:88) λ (cid:96) n c λ ( t ) e λ . Then (cid:88) λ (cid:96) nl ( λ )= k c λ ( t ) = (cid:88) ¯ a ∈ AO k ( G ) t asc −→ G (¯ a ) where AO k ( G ) is the set of acyclic orientations of G with k sinks and asc −→ G (¯ a ) is thenumber of edges of G for which ¯ a and −→ G have the same orientation.The simplest proper circular arc digraphs that are not also unit interval digraphsare the cycles on n vertices, −→ C n := −→ G ∗ n, . In [Sta95, Proposition 5.4], Stanley provides aformula for the e -basis expansion of the chromatic symmetric functions of undirected This is shown to be a palindromic polynomial in [SW16]. An equivalent form of this conjecture is also noted in [AP]. This is shown to be a palindromic polynomial in [Ell].
HROMATIC QUASISYMMETRIC FUNCTIONS OF DIRECTED GRAPHS 9 cycles that show that they are e -positive. We refine his formula for the chromaticquasisymmetric functions of directed cycles. Theorem 12.
For directed cycles, −→ C n , (cid:88) n ≥ X −→ C n ( x , t ) z n = t (cid:88) k ≥ k [ k − t e k z k − t (cid:88) k ≥ [ k − t e k z k where [ k ] t = 1 + t + · · · + t k − . Consequently, X −→ C n ( x , t ) is palindromic, e -positive, and e -unimodal. We prove this by extending the technique of Stanley for the t = 1 case, which usesthe transfer matrix method [Sta12], and by using a result on permutation statistics byMantaci and Rakotondrajao [MR03].The following two propositions give combinatorial interpretations for the coefficientsin the e -basis expansion of the chromatic quasisymmetric functions of −→ P n , the path on nvertices oriented in one direction, and of −→ C n , the cycle on n vertices oriented cyclically.The first proposition uses our generating function for the e -basis expansion of X −→ C n ( x , t )seen in Theorem 12. The second proposition uses a similar generating function byShareshian and Wachs [SW16] for the e -basis expansion of X −→ P n ( x , t ) . Though the proofsof these propositions rely on formulas that already give us e -positivity, perhaps thereis a generalization of these interpretations that suggests a possible method for proving e -positivity for larger classes of graphs. Proposition 13.
Let −→ C n be the cycle on n vertices oriented cyclically with C n as itsunderlying undirected graph and let X −→ C n ( x , t ) = (cid:88) λ (cid:96) n c λ ( t ) e λ . Then c λ ( t ) = (cid:88) ¯ a ∈ AO λ ( C n ) t asc −→ Cn (¯ a ) where AO λ is the set of all acyclic orientations of C n such that the number of verticesbetween consecutive sinks of ¯ a is λ − , λ − , ..., λ k − −→ C n (¯ a ) isthe number of edges of −→ C n for which the orientations of −→ C n and ¯ a agree. Example 14.
Suppose we have the following acyclic orientation of C , the underlyingundirected graph of −→ C . For convenience, we label the vertices with the elements of [9]such that the edges of the original −→ C are oriented from smaller label to larger label,except for the edge between 1 and 9, i.e. 1 → → · · · → → . An alternative proof of this result was subsequently obtained in [AP]. These two propositions were obtained independently in [AP].
This corresponds to a t e term, because there are 3 vertices between the sinks 2and 6, 1 vertex between the sinks 6 and 8, and 2 vertices between the sinks 8 and 2.The red edges are the 3 ascents of this orientation. Proposition 15.
Let −→ P n be the path of length n with edges oriented in one directionwith P n as its underlying undirected graph and let X −→ P n ( x , t ) = (cid:88) λ (cid:96) n c λ ( t ) e λ . Then c λ ( t ) = (cid:88) ¯ a ∈ AO λ ( P n ) t asc −→ Pn (¯ a ) where AO λ ( P n ) is the set of all acyclic orientations of P n such that the number ofvertices between consecutive sinks of ¯ a (including the total number of vertices beforethe first sink and after the last sink) is λ − , λ − , ..., λ k − −→ P n (¯ a )is the number of edges of −→ P n for which the orientations of −→ P n and ¯ a agree. Example 16.
Suppose we have the following acyclic orientation of P , the underlyingundirected graph of −→ P . For convenience, we label the vertices with the elements of [8]such that the edges of the original digraph −→ P are oriented from smaller label to largerlabel, i.e. 1 → → · · · → t e term, because there are 3 vertices between the sinks 2and 6, 1 vertex between the sinks 6 and 8, 1 vertex before the 2 and none after the 8.The red edges are the 4 ascents of this orientation. Acknowledgements
I would like to thank Richard Stanley for his helpful suggestion that made this workpossible. I would also like to thank my advisor, Michelle Wachs, for all of her guidanceand encouragement.
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