LLT polynomials, elementary symmetric functions and melting lollipops
LLLT POLYNOMIALS, ELEMENTARY SYMMETRIC FUNCTIONSAND MELTING LOLLIPOPS
PER ALEXANDERSSON
Abstract.
We conjecture an explicit positive combinatorial formula for theexpansion of unicellular LLT polynomials in the elementary symmetric basis.This is an analogue of the Shareshian–Wachs conjecture previously studied byPanova and the author in 2018. We show that the conjecture for unicellularLLT polynomials implies a similar formula for vertical-strip LLT polynomials.We prove positivity in the elementary basis in for the class of graphs called“melting lollipops” previously considered by Huh, Nam and Yoo. This isdone by proving a curious relationship between a generalization of charge andorientations of unit-interval graphs.We also provide short bijective proofs of Lee’s three-term recurrences forunicellular LLT polynomials and we show that these recurrences are enough togenerate all unicellular LLT polynomials associated with abelian area sequences.
Contents
1. Introduction 12. Preliminaries 33. Recursive properties of LLT polynomials 84. Recursions for the conjectured formula 135. The Hall–Littlewood case 176. Generalized cocharge and e-positivity 197. A possible approach to settle the main conjecture 21References 231.
Introduction
Background on LLT polynomials.
LLT polynomials were introduced byLascoux, Leclerc and Thibon in [LLT97], and are q -deformations of products ofskew Schur functions. An alternative combinatorial model for the LLT polynomialswas later introduced in [HHL05a] while studying Macdonald polynomials. In theirpaper, LLT polynomials are indexed by a k -tuple of skew shapes. In the case eachsuch skew shape is a single box, the LLT polynomial is said to be unicellular LLTpolynomial . Such unicellular LLT polynomials are the main topic of this paper.1.2. Background on chromatic symmetric functions.
In [CM17] Carlssonand Mellit introduced a more convenient combinatorial model for unicellular LLT a r X i v : . [ m a t h . C O ] N ov PER ALEXANDERSSON polynomials, indexed by (area sequences of) Dyck paths. They also highlighted animportant relationship using plethysm between unicellular LLT polynomials andthe chromatic quasisymmetric functions introduced by Shareshian and Wachs in[SW12].The chromatic quasisymmetric functions refine the chromatic symmetric functionsintroduced by Stanley in [Sta95]. The Stanley–Stembridge conjecture [SS93] statesthat such chromatic symmetric functions associated with unit interval graphs , andmore generally, incomparability graphs of 3 + 1-free posets are positive in theelementary symmetric basis, or e-positive for short. Their conjecture was refinedwith the introduction of an additional parameter q in [SW12]. The class of graphsfor which this conjecture is believed to hold was later extended to the class of circular unit interval graphs in [Ell17a, Ell17b] where it is conjectured that thechromatic quasisymmetric functions expanded in the e-basis have coefficients in N [ q ], see Conjecture 13 below. To this date, there is still not even a conjecturedcombinatorial formula for the e-expansion of the chromatic symmetric functions.The idea of studying LLT polynomials in parallel with quasisymmetric chromaticfunctions originated in [CM17], althought the connection is perhaps in hindsightapparent in the techniques used in [HHL05a]. We also mention an interesting paperby Haglund and Wilson [HW17] explores the connection between the integral-formMacdonald polynomials and the quasisymmetric chromatic functions.1.3. Main results.
In [AP18], we stated an analogue of the Shareshian–Wachsconjecture regarding e-positivity of unicellular LLT polynomials, G a ( x ; q + 1) andproved the conjecture in a few cases. We also provided many similarities betweenunicellular LLT polynomials and chromatic quasisymmetric functions associatedwith unit-interval graphs. The problem of e-positivity of unicellular LLT polynomialsis the main topic of this article.The main results are: • We present a precise conjectured combinatorial formula for the e-expansionof G a ( x ; q + 1). Our conjecture states that the unicellular LLT polynomialG a ( x ; q ) is given asG a ( x ; q + 1) := X θ ∈ O ( a ) q asc( θ ) e π ( θ ) ( x ) . (1)where O ( a ) is the set of orientations of the unit interval graph with areasequence a , and π ( θ ) is an explicit partition-valued statistic on such orien-tation. This formula can be extended to vertical-strip LLT polynomials,and has been verified on the computer for all unit-interval graphs up to 10vertices. This formula is surprising, as there is still no analogous conjecturedformula for chromatic symmetric functions.A possible application of (1) is to find a positive combinatorial formulafor the Schur-expansion of G a ( x ; q ). • We prove in Corollary 31 that the conjectured formula (1) implies a general-ized formula for the so called vertical-strip LLT polynomials . Furthermore,we prove that (1) holds for the family of complete graphs and line graphs. • Analogous recursions for the unicellular LLT polynomials are given by Leein [Lee18]. We give short bijective proofs of these recurrences and show that
LT POLYNOMIALS, ELEMENTARY SYMMETRIC FUNCTIONS AND MELTING LOLLIPOPS3 all graphs associated with abelian Hessenberg varieties can be computedrecursively via Lee’s recurrences, starting from unicellular LLT polynomialsassociated with the complete graphs. • In Section 5, we prove that the transformed Hall–Littlewood polynomialsH λ ( x ; q + 1) are positive in the complete homogeneous basis. This impliesthat a corresponding family of vertical-strip LLT polynomials are e-positive.Note that vertical-strip LLT polynomials appear in diagonal harmonics,see for example [Ber17, Section 4] and [HHL + • In Section 6, we prove a curious identity between a generalization of charge,denoted wt a ( T ), and the set of orientations, O ( a ), of a unit-interval graphΓ a . It states that X λ ‘ n X T ∈ SYT( λ ) ( q + 1) wt a ( T ) s λ ( x ) = X θ ∈ O ( a ) q asc( θ ) e σ ( θ ) ( x ) , (2)where asc( · ) and σ ( · ) are certain combinatorial statistics on orientations.This version of charge was considered in [HNY18] in order to prove Schurpositivity for unicellular LLT polynomials in the melting lollipop graph case.As a consequence, we get an explicit positive e-expansion the case of melting lollipop graphs which has previously been considered in [HNY18].The corresponding family of chromatic quasisymmetric functions was con-sidered in [Dv18] where they were proved to be e-positive. Note howeverthat the statistic π ( θ ) in (1) and σ ( θ ) in (2) are different.The paper is organized as follows. We first introduce the family of unicellular–and vertical-strip LLT polynomials and some of their basic properties. In Section 3,we prove several recursive identities for such LLT polynomials. In particular, weshow that the recursions by Lee [Lee18] can be used to construct unicellular LLTpolynomials indexed by any abelian area sequence.Some vertical-strip LLT polynomials are closely related to the transformed Hall–Littlewood polynomials. In Section 5, we show that the transformed Hall–Littlewoodpolynomials H λ ( x ; q + 1) are h-positive, which gives further support for the mainconjecture.In Section 6, we study the relationship between a type of generalized cochargeintroduced in [HNY18] and e-positivity. This provides a proof that unicellular LLTpolynomials given by melting lollipop graphs are e-positive.Finally in Section 7, we describe a possible approach to prove (1) by a comparisonin the power-sum symmetric basis.2. Preliminaries
We use the same notation and terminology as in [AP18]. The reader is assumed tohave a basic background on symmetric functions and related combinatorial objects,see [Sta01, Mac95]. All Young diagrams and tableaux are presented in the Englishconvention.
PER ALEXANDERSSON
Dyck paths and unit-interval graphs. An area sequence is an integer vector a = ( a , . . . , a n ) which satisfies • ≤ a i ≤ i − ≤ i ≤ n and • a i +1 ≤ a i + 1 for 1 ≤ i < n .The number of such area sequences of size n is given by the Catalan numbers. Notethat [HNY18] uses a reversed indexing of entries in area sequences. Definition 1.
For every area sequence of length n , we define a unit interval graph Γ a with vertex set [ n ] and the directed edges( i − a i ) → i, ( i − a i + 1) → i, ( i − a i + 2) → i, . . . , ( i − → i (3)for all i = 1 , . . . , n . We say that ( u, v ) with u < v is an outer corner of Γ a if ( u, v )is not an edge of Γ a , and either • u + 1 = v or • ( u + 1 , v ) and ( u, v −
1) are edges of Γ a . Example 2.
We can illustrate area sequences and their corresponding unit-intervalgraphs as
Dyck diagrams , as is done in [Hag07, AP18]. For example, (0 , , , , ,
46 56
35 45
14 24 34
13 23
21 (4)where the area sequence specify the number of white squares in each row, bottom totop. The squares on the main diagonal are the vertices of Γ a , and each white squarecorrespond to a directed edge of Γ a . In the second figure we see this correspondencewhere edge ( i, j ) is marked as ij . The outer corners of Γ a are (2 ,
5) and (3 , Caution:
We do not really distinguish the terms area sequence , Dyck diagram and unit interval graph , as they all relate to the same objects. What term is useddepends on context and what features we wish to emphasize.Let Γ a be an unit interval graph with n vertices. We let a T denote the areasequence of Γ a where all edges have been reversed, and every vertex j ∈ [ n ] hasbeen relabeled with n + 1 − j . This operation corresponds to simply transposingthe Dyck diagram. Lemma 3 (See [AP18]) . The entries in an area sequence a is a rearrangement ofthe entries in a T . Most results in this paper concerns a few special classes of area sequences.
Definition 4.
An area sequence of length n is called rectangular if either a =(0 , , , . . . , n −
1) or there is some k ∈ [ n ] such that a i = i − i = 1 , , . . . , k and a j = j − k − j = k + 1 , k + 2 , . . . , n. LT POLYNOMIALS, ELEMENTARY SYMMETRIC FUNCTIONS AND MELTING LOLLIPOPS5
This condition is equivalent with all non-edges forming a k × ( n − k )-rectangle in theDyck diagram. Furthermore, an area sequence a is called abelian whenever a i ≥ a i for some rectangular sequence a . For example, the area sequence in (4) is abelian.The terminology is motivated by [HP17], where abelian area sequences areassociated with abelian Hessenberg varieties.We will also consider the following families of area sequences: • The complete graphs, (0 , , , . . . , n − • The line graphs (0 , , , . . . , • Lollipop graphs, where a i = ( i − i = 1 , . . . , m i = m + 1 , . . . , m + n for some m, n ≥ • Melting complete graph, a i = ( i − i = 1 , , . . . , n − n − k − i = n where 0 ≤ k ≤ n − • Melting lollipop graphs, defined as a i = i − i = 1 , . . . , m − m − − k for i = m i = m + 1 , . . . , m + n for m, n ≥ ≤ k ≤ m − Vertical strip diagrams. A vertical strip diagram is a Dyck diagram wheresome of the outer corners have been marked with → . We call such an outer cornera strict edge . These markings correspond to some extra oriented edges of Γ a . Weuse the notation Γ a , s to denote a directed graph with some additional strict edges s and refer to the graph Γ a , s as a vertical strip diagram as well. Example 5.
Below is an example of a vertical strip diagram. → → ,
4) and (3 ,
6) are strict of Γ a , s , and the directed edges of Γ a (which arealso edges of Γ a , s ) are { (1 , , (1 , , (2 , , (2 , , (3 , , (3 , , (4 , , (4 , , (5 , } . Note that this is another example of a diagram with an abelian area sequence.
PER ALEXANDERSSON
Vertical strip LLT polynomials.
Let Γ a , s be a vertical strip diagram. A valid coloring κ : V (Γ a , s ) → N is a vertex coloring of Γ a , s such that κ ( u ) < κ ( v )whenever ( u, v ) is a strict edge in s . Given a coloring κ , an ascent of κ is a (directed)edge ( u, v ) in Γ a , s such that κ ( u ) < κ ( v ). Note that strict edges do not count asascents. Let asc( κ ) denote the number of ascents of κ . Definition 6.
Let Γ a , s be a vertical strip diagram. The vertical strip LLT polynomial G a , s ( x ; q ) is defined as G a , s ( x ; q ) := X κ : V (Γ a , s ) → N x κ q asc( κ ) (5)where the sum is over valid colorings of Γ a , s . Whenever s = ∅ , we simply writeG a ( x ; q ) and refer to this as a unicellular LLT polynomial .As an example, here is G ( x ; q ) expanded in the Schur basis:G ( x ; q ) = q s + ( q + q + q )s + ( q + q )s + (1 + q + q )s + s . The polynomials G a , s ( x ; q ) are known to be symmetric, and correspond to classicalLLT polynomials indexed by k -tuples of skew shapes as in [HHL05a]. In fact, theunicellular LLT polynomials correspond to the case when all shapes in the k -tupleare single cells, and the vertical strip case correspond to k -tuples of single columns.This correspondence is proved in [AP18] and is also done implicitly in [CM17]. Thereis a close connection with the ζ map used by Haglund and Loehr, see [HL05, Hag07]. Example 7.
In the following vertical strip diagram, we illustrate a valid coloring κ where we have written κ ( i ) on vertex i . That is, κ (1) = 1, κ (2) = 3, κ (3) = 2, etc. → → → → → → → κ ) have been marked with → . Hence, κ contributes with q x x x x to the sum in (5).2.4. A conjectured formula.Definition 8.
Let a be an area sequence of length n and s be some strict edges ofΓ a . Let O ( a , s ) denote the set of orientations of the graph Γ a (seen as an undirectedgraph) together with the extra directed edges in s . Thus, the cardinality of O ( a , s )is simply 2 a + ··· + a n . If s = ∅ , we simply write O ( a ) for the set of orientations of Γ a .Given θ ∈ O ( a , s ), an edge ( u, v ) is an ascending edge in θ if it is oriented in thesame manner as in Γ a . Let asc( θ ) denote the number of ascending edges in θ . Notethat edges in s are not considered to be ascending! We now define the highest reachable vertex , hrv θ ( u ) for u ∈ [ n ] as the maximal v such that there is a path from u to v in θ using only strict and ascending edges .Note that hrv θ ( u ) ≥ u for all u . The orientation θ defines a set partition π ( θ ) ofthe vertices of Γ a , where two vertices are in the same part if and only if they have LT POLYNOMIALS, ELEMENTARY SYMMETRIC FUNCTIONS AND MELTING LOLLIPOPS7 the same highest reachable vertex. Let π ( θ ) denote the partition given by the sizesof the sets in π ( θ ).Let a be an area sequence and s be some strict edges of Γ a . Define the symmetricfunction ˆG a , s ( x ; q ) via the relationˆG a , s ( x ; q + 1) := X θ ∈ O ( a , s ) q asc( θ ) e π ( θ ) ( x ) . (6) Example 9.
Below, we illustrate an orientation θ ∈ O ( a , s ), where a = (0 , , , , , s = { (1 , , (2 , } . As before, strict edges and edges contributing to asc( θ ) aremarked with → . → → → → → → → θ (2) = hrv θ (5) = hrv θ (6) = 6 and hrv θ (1) = hrv θ (3) = hrv θ (4) = 4.Thus π ( θ ) = { , } and the orientation θ contributes with q e ( x ) in 6. Thefull polynomial ˆG a , s ( x ; q + 1) is(4 q + 20 q + 41 q + 44 q + 26 q + 8 q + q )e + (2 q + 7 q + 9 q + 5 q + q )e +(2 q + 9 q + 16 q + 14 q + 6 q + q )e +(4 q + 22 q + 48 q + 53 q + 31 q + 9 q + q )e +(4 q + 14 q + 18 q + 10 q + 2 q )e + ( q + 8 q + 20 q + 22 q + 11 q + 2 q )e +(1 + 3 q + 3 q + q )e + ( q + 3 q + 3 q + q )e Conjecture 10 (Main conjecture) . For any vertical-strip LLT polynomial G a , s ( x ; q ) we have that G a , s ( x ; q ) = ˆG a , s ( x ; q ) . Note that this conjecture implies that G a , s ( x ; q + 1) is e-positive, with theexpansion given as a sum over all orientations of Γ a . Such a conjecture was firststated in [AP18] but without a precise definition of π ( θ ). Conjecture 10 is a naturalanalogue of the Shareshian–Wachs conjecture, [SW12, SW16], and therefore is alsoclosely related to the Stanley–Stembridge conjecture [SS93, Sta95]. There is alsoa natural generalization of Equation (6) that predicts the e-expansion of the LLTpolynomials indexed by circular area sequences considered in [AP18].2.5. Properties of LLT polynomials.
We use standard notation and let ω be theinvolution on symmetric functions that sends the complete homogeneous symmetricfunction h λ to the elementary symmetric function e λ , or equivalently, sends s λ tos λ . Proposition 11 (See [AP18]) . For any area sequence a of length n , ω G a ( x ; q ) = q a + a + ··· + a n G a T ( x ; 1 /q ) (7) where a T denotes the transpose of the Dyck diagram. PER ALEXANDERSSON
In [AP18], we gave a proof that ω G a , s ( x ; q + 1) is positive in the power-sum basis.It also follows from a much more general result given in [AS19]. Note that if f ( x ) ise-positive, then ωf ( x ) is positive in the power-sum basis. Later in Proposition 48,the power-sum expansion of ω G a , s ( x ; q + 1) is stated explicitly.The following lemma connects the LLT polynomials with the chromatic quasisym-metric functions X a ( x ; q ) introduced in [SW12]. The function X a ( x ; q ) is definedexactly as G a ( x ; q ) but the sum in Equation (5) is taken only over proper coloringsof Γ a , so that no monochromatic edges are allowed. Lemma 12 (Adaptation of [CM17, Prop. 3.5]. See also [HHL05a, Sec. 5.1]) . Let a be a Dyck diagram. Then ( q − − n G a [ x ( q − q ] = X a ( x ; q ) , (8) where the bracket denotes a substitution using plethysm. From this formula, together with Conjecture 10, we have a novel conjecturedformula for the chromatic quasisymmetric functions:X a ( x ; q ) = X θ ∈ O ( a ) ( q − asc( θ ) e π ( θ ) [ x ( q − q − n . (9)Perhaps it is possible to do some sign-reversing involution together with plethysmmanipulations to obtain the e-expansion of X a ( x ; q ) and thus find a candidateformula for the Shareshian–Wachs conjecture. Conjecture 13 (Shareshian–Wachs [SW12, SW16]) . There is some partition-valuedstatistic ρ on acyclic orientations of Γ a , such that X a ( x ; q ) = X θ ∈ AO ( a ) q asc( θ ) e ρ ( θ ) ( x ) . Here AO ( a ) denotes the set of acyclic orientations of Γ a . Note that the original Stanley–Stembridge conjecture is closely related to the q = 1 case, which was stated for the incomparability graphs of 3 + 1-avoiding posets. Problem 14.
Prove that the family ˆG a ( x ; q ) defined in (6) fulfills the involutionidentity (7). 3. Recursive properties of LLT polynomials
We shall now cover several recursive relations for the vertical-strip LLT poly-nomials. Our proofs are bijective and directly use the combinatorial definition asa weighted sum over vertex colorings. We illustrate these bijections with Dyckdiagrams where only the relevant vertices and edges are shown.The reader thus is encouraged to interpret a diagram as a weighted sum overcolorings, where decorations of the diagrams indicate restrictions of the colorings, orhow the colorings contribute to asc( · ). For example, given an edge (cid:15) of Γ a , s , thereare two possible cases. Either (cid:15) contributes to the number of ascents, or it does not.We can illustrate this simply as = ↓ + q → LT POLYNOMIALS, ELEMENTARY SYMMETRIC FUNCTIONS AND MELTING LOLLIPOPS9 where the white box is the edge (cid:15) and ↓ indicates an edge that cannot be anascent. Note that the vertices shown do not need to have consecutive labels — theintermediate vertices (and edges) are simply not shown. Shaded boxes are not edgesof Γ a and therefore does not contribute to ascents of the coloring. To conclude, theclass of diagrams considered here may be described as follows: • The white boxes are determined by some area sequence a , so that eachwhite box is an edge in Γ a . • Every edge (box) is either white or shaded. • Only white boxes contribute to the ascent statistic. • A box (white or shaded) may contain an arrow, a → or ↓ , imposing a strictor weak inequality requirement, respectively, on the colorings. In particular,a white box containing a → is thus a sum over colorings where this particularedge must be an ascent.Note that this is a slightly broader class of diagrams than the class of vertical-stripdiagrams, as the additional arrows impose more restrictions on the colorings.The following recursive relationship allows us to express vertical-strip LLT polyno-mials as linear combinations of unicellular LLT polynomials. Later in Proposition 30,we prove that the polynomials in Equation (6) satisfy the same recursion. We usethe notation a ∪ { (cid:15) } to describe the area sequence of the unit interval graph wherethe edge (cid:15) has been added to the edges of Γ a . The notation s ∪ { (cid:15) } for strict edgesis interpreted in a similar manner. Proposition 15. If Γ a , s is a vertical strip diagram, and (cid:15) is a non-strict outercorner of Γ a , s , then G a ∪{ (cid:15) } , s ( x ; q + 1) = G a , s ( x ; q + 1) + q G a , s ∪{ (cid:15) } ( x ; q + 1) . (10) Proof.
By shifting the variable q , the identity can be restated asG a ∪{ (cid:15) } , s ( x ; q ) + G a , s ∪{ (cid:15) } ( x ; q ) = q G a , s ∪{ (cid:15) } ( x ; q ) + G a , s ( x ; q ) , (11)which in (as sum over colorings) diagram form can be expressed as follows. The twovertices shown are the vertices of (cid:15) .+ → = q → +The first and last diagram can be expanded into subcases, (cid:16) ↓ + q → (cid:17) + → = q → + (cid:16) ↓ + → (cid:17) and here it is evident that both sides agree. (cid:3) The above recursion seem to relate to certain recursions on Catalan symmetricfunctions, see [BMPS18, Prop. 5.6]. Catalan symmetric functions are very similarin nature to LLT polynomials.3.1.
Lee’s recursion.
In Proposition 18 below, we prove a recursion on certainLLT polynomials. We then show that this relation is equivalent to Lee’s recursion,given in [Lee18, Thm 3.4].
Definition 16.
Let a be an area sequence of length n ≥
3. An edge ( i, j ) ∈ E (Γ a ),3 ≤ j ≤ n , is said to be admissible if the following four conditions hold: • i = j − a j • j = n or a j ≥ a j +1 + 1 • a j ≥ • a i + 1 = a i +1 .The last condition is automatically satisfied if the first three are true and a is abelian.Note that if ( i, j ) is admissible, then for all k < i or k > i + 1 we have( k, i ) ∈ E (Γ a ) ⇔ ( k, i + 1) ∈ E (Γ a ) and ( i, k ) ∈ E (Γ a ) ⇔ ( i + 1 , k ) ∈ E (Γ a ) . (12)These properties are crucial in later proofs. Example 17.
For the following diagram a , the edge (2 ,
5) is admissible.654321Let e j denote the j th unit vector. Proposition 18.
Suppose ( i, j ) is an admissible edge of the area sequence a , set a := a − e j and a := a − e j and s := { ( i, j ) } , s := { ( i + 1 , j ) } . Then G a , s ( x ; q ) = q G a , s ( x ; q ) . (13) Proof.
We use the diagram-type proof as before, now only showing the vertices i , i + 1 and j . The identity we wish to show is then presented as → = q → . Both sides are considered as a weighted sum over colorings with restrictions indicatedby → . Subdividing these sums into subcases by forcing additional inequalities gives q →→ + →↓ = q (cid:18) →→ + ↓→ (cid:19) . Two terms cancel and additional inequalities follows by transitivity. It thereforesuffices to prove the following. q →↓→ = q ↓→↓ Note that the additional q in the left hand side appears due to the ascent ( i, i + 1).There is now a simple q -weight-preserving bijection between colorings on thediagram on the left hand side, and colorings of the diagram on the right hand side.For a coloring κ in the left hand side, we have κ ( i ) < κ ( j ) ≤ κ ( i + 1), while on theright hand side, we have κ ( i + 1) < κ ( j ) ≤ κ ( i ). Hence, vertex i and vertex i + 1are never assigned the same color.The bijection is to simply swap the colors of the adjacent vertices i and i + 1. Theproperty in Equation (12) ensures that the number of ascending edges are preservedunder this swap. (cid:3) LT POLYNOMIALS, ELEMENTARY SYMMETRIC FUNCTIONS AND MELTING LOLLIPOPS11
Corollary 19 (Local linear relation [Lee18, Thm 3.4]) . Let a be an area sequencefor which ( i, j ) is admissible, and let a := a , a := a − e j and a := a − e j . Then G a ( x ; q ) − G a ( x ; q ) = q (G a ( x ; q ) − G a ( x ; q )) . (14) Proof.
We see that the left hand side of (14) can be rewritten in diagram form usingEquation (10): LHS = − = ( q − → The right hand side is treated in a similar manner:RHS = q (cid:18) − (cid:19) = q ( q − → The identity in (13) now implies that LHS = RHS. (cid:3)
Example 20.
As an illustration of Corollary 19, we have ( i, j ) = (2 ,
5) and thefollowing three Dyck diagrams. a = 654321 a = 654321 a = 6543213.2. The dual Lee recursion.
There is a “dual” version of Corollary 19, obtainedby applying ω to both sides of (14), and then apply the relation in (7). We shallnow state this in more detail. Definition 21.
Let a be an area sequence of length n ≥
3. An edge ( i, j ) is said tobe dual-admissible if the edge ( n + 1 − j, n + 1 − i ) is admissible for a T .We can then formulate the dual versions of Proposition 18 and Corollary 19. Proposition 22 (The dual Lee recursion) . Let a be an area sequence for which ( i, j ) is dual-admissible and let a := a , a := a − e j and a := a − e j − e j − . Then G a , s ( x ; q ) = q G a , s ( x ; q ) (15) and G a ( x ; q ) − G a ( x ; q ) = q (G a ( x ; q ) − G a ( x ; q )) (16) where s := { ( i, j ) } and s := { ( i, j − } .Proof sketch. We can either prove these identities by applying ω as outlined above,or bijectively using diagrams. We leave out the details. (cid:3) Example 23.
Proposition 18 applies in the following generic situation. Here, theedge ( x, z ) is an admissible edge. The crucial condition in (12) states that the area of the rows with vertices x and y in the diagram differ by exactly one.( a , s ) = → x y z ( a , s ) = → x y z (17)Similarly, the dual recursion in Equation (15) applies in the following situation,where ( x, z ) is a dual-admissible edge:( a , s ) = → zyx ( a , s ) = → zyx (18)3.3. Recursion in the complete graph case.
We end this section by recallinga recursion for LLT polynomials in the complete graph case.
Proposition 24 ([AP18, Prop.5.8]) . Let G K n ( x ; q ) denote the LLT polynomial forthe complete graph, where the area sequence is (0 , , , . . . , n − . Then G K n ( x ; q ) = n − X i =0 G K i ( x ; q ) e n − i ( x ) n − Y k = i +1 (cid:2) q k − (cid:3) , G K ( x ; q ) = 1 . (19) Lemma 25. If a is rectangular and the non-edges form a k × ( n − k ) -rectangle inthe Dyck diagram, then G a ( x ; q ) = G K k ( x ; q )G K n − k ( x ; q ) .Proof. The unit-interval graph Γ a is a disjoint union of two smaller completegraphs, so this now follows immediately from the definition of unicellular LLTpolynomials. (cid:3) For the remaining of this section, it will be more convenient to use the notationin [Lee18], and index unicellular LLT polynomials of degree n with partitions λ thatfit inside the staircase ( n − , n − , . . . , , , n and let the area sequence a correspond to the partition λ where λ i = n − i − a n +1 − i . Hence, λ is exactly theshape of the (shaded) non-edges in the Dyck diagram. By definition, λ is abelian ifit fits inside some k × ( n − k )-rectangle. Lemma 26 (Follows from [HNY18, Thm. 3.4]) . Let λ be abelian with ‘ ≥ partssuch that λ ‘ < λ ‘ − . Let µ = ( λ , λ , . . . , λ ‘ − ) and ν = ( λ , λ , . . . , λ ‘ − , λ ‘ + 1) . Then there are rational functions c ( q ) and d ( q ) such that G λ ( x ; q ) = c ( q )G µ ( x ; q ) + d ( q )G ν ( x ; q ) .Proof. Use Corollary 19 repeatedly on row ‘ of µ . (cid:3) LT POLYNOMIALS, ELEMENTARY SYMMETRIC FUNCTIONS AND MELTING LOLLIPOPS13
Example 27.
To illustrate Lemma 26, we have the following three partitions: λ = µ = , ν = Proposition 28.
Every G λ ( x ; q ) where λ is abelian, can be expressed recursivelyvia Lee’s recurrences, as a linear combination of some G µ j ( x ; q ) where the µ j arerectangular.Proof. Let λ be abelian partition with exactly ‘ parts, so that it fits in a ‘ × ( n − ‘ )-rectangle. We shall do a proof by induction over λ , and in particular its number ofparts.(1) Case λ = ∅ . This is rectangular by definition.(2)
Case λ = ( n − . This is rectangular.(3)
Case ‘ = 1 . Use Lemma 26 to reduce to Case (1) and Case (2).(4)
Case ‘ > and λ i ≤ ‘ − i for some i ∈ [ ‘ ] . The conditions imply thatit is possible to remove a 2 × × λ and obtain anew partition. Hence we can use Lee’s recursion to express G λ ( x ; q ) usingpolynomials indexed by two smaller partitions. For example, this caseapplies in the following situation: λ = −→ and (20)(5) Case ‘ > and λ i > ‘ − i for all i ∈ [ ‘ ] . Three things can happen here,and it is easy to see that this list is exhaustive. • λ is rectangular and we are done. • We can add a 2 × × λ without increasing ‘ andstill obtain a partition. Similar to Case (4), we can therefore reduce tocases where | λ | has increased by 1 and 2. • Lemma 26 can be applied, thus reducing λ to a case where ‘ has strictlybeen decreased, and a case where λ has increased by one box.Notice that Case (4) reduces only back to Case (4), or a case where ‘ is decreased,and the same goes for Case (5). There are therefore no circular dependenciesamongst these cases and the induction is valid. (cid:3) Recursions for the conjectured formula
In this section, we prove that ˆG( x ; q ) also fulfills the recursion in Proposition 15.We use similar bijective technique as in Section 3, but diagrams now representweighted sums over orientations as in Equation (6). Note that we now also considerthe shifted polynomial ˆG a , s ( x ; q + 1).Each diagram now represents a weighted sum over orientations, where the weightof a single orientation θ is q asc( θ ) e π ( θ ) . The class of diagrams we now consider is asfollows. • The white boxes are determined by some area sequence. • Every edge (box) is either white or shaded. • Only white boxes contribute to the ascent statistic. • A box (white or shaded) may contain an arrow, a → or ↓ , imposing arestriction on the orientations being summed over. In particular, a white box containing a → is thus a sum over orientations where this particularedge must be an ascent.Hence, each diagram represents a sum over 2 W orientations, where W is the numberof white boxes not containing any arrow. Example 29.
Suppose the following diagram illustrates the entire graph. Thediagram represents the weighted sum over all orientations of the non-specified edges( x, y ) and ( y, z ). The edge ( x, z ) is strict, and ( z, w ) is forced to be ascending.Remember that each ascending edge contributes with a q -factor. → w → zyx There are four orientations in total, → w → ↓ z ↓ yx → w → ↓ z → yx → w →→ z ↓ yx → w →→ z → yx which according to (6) give the sum q e + q e + q e + q e .In the diagrams below, only relevant vertices of the graphs are included. Proposition 30. If Γ a , s is a vertical-strip graph, with (cid:15) being a non-strict outercorner, then ˆG a ∪{ (cid:15) } , s ( x ; q + 1) = q ˆG a , s ∪{ (cid:15) } ( x ; q + 1) + ˆG a , s ( x ; q + 1) . (21) Proof.
In diagram form, this amounts to showing that orientations of the diagramin the left hand side can be put in q -weight-preserving bijection with the disjointsets of orientations in the right hand side, while also preserving the π ( · )-statistic.Thus we wish to prove that yx = q → yx + yx . Consider an orientation in the left hand side. There are two cases to consider: • The edge ( x, y ) is ascending. We map the orientation to an orientation ofthe first diagram in the right hand side, by preserving the orientation of allother edges. • The edge ( x, y ) is non-ascending. We map this to the second diagram, bypreserving the orientation of all other edges.In both cases above, note that both the q -weight and π ( · ) is preserved under thismap. (cid:3) Corollary 31.
If Conjecture 10 holds in the unit-interval case, it holds in thevertical-strip case.Proof.
We can use Proposition 30 and Proposition 15 to recursively remove all strictedges. Since both families satisfy the same recursion, we have that the unicellularcase of Conjecture 10 implies the vertical-strip case. (cid:3)
LT POLYNOMIALS, ELEMENTARY SYMMETRIC FUNCTIONS AND MELTING LOLLIPOPS15
The complete graph recursion and line graphs.
Analogous to Proposi-tion 24, we have a recursion for the corresponding ˆG K n ( x ; q ), where we again considerthe complete graph case. Here, (cid:0) [ n ] k (cid:1) denotes the set of k -subsets of { , . . . , n } . Lemma 32.
The polynomial ˆG K n ( x ; q ) satisfy ˆG K ( x ; q ) := 1 and ˆG K n ( x ; q + 1) is equal to n − X i =0 ˆG K i ( x ; q + 1)e n − i ( x ) X S ∈ ( [ n − n − − i ) n − − i Y j =1 ( q + 1) s j − j [( q + 1) j − . (22) Proof.
We first give an argument for the recursion in (22). Given an orientation θ of K i , we can construct a new orientation θ of K n by inserting a new part ofsize n − i in the vertex partition where vertex n is a member. Choose an i -subsetof [ n −
1] and let θ define the orientation of the edges in θ on these vertices. Theremaining n − i − S = { s , . . . , s n − i − } where n > s > s > · · · > s n − i − ≥
1. Each element s j musthave at least one ascending edge to either vertex n , or to another member in S larger than s j , but all other choices of ascending edges are allowed. It then followsthat that for such a subset S , there are n − − i Y j =1 ( q + 1) n − s j − j [( q + 1) j − · )-weighted ways of choosing subsets of ascending edges in θ so that all verticesin S has n as highest reachable vertex. Hence, X S ∈ ( [ n − n − − i ) n − − i Y j =1 ( q + 1) n − s j − j [( q + 1) j − · )-weighted count of the number of orientations of K n , where the part ofthe vertex-partition containing n has exactly n − i members. Finally, by sendingeach s i to n − s i , which is an involution on (cid:0) [ n − n − − i (cid:1) , we get the desired formula. (cid:3) We shall now prove that ˆG K n ( x ; q ) = G K n ( x ; q ). By using Lemma 32 andProposition 24, this follows from the following lemma. Lemma 33.
For all n and ≤ i ≤ n − , we have that n − Y k = i +1 (cid:2) q k − (cid:3) = X S ∈ ( [ n ] n − i − ) n − − i Y j =1 q s j − j [ q j − . Proof.
A small rewrite in each of the product indices gives ( n − i − Y k =1 (cid:2) q k +1 − (cid:3) = X S ∈ ( [ n ]( n − i − ) ( n − i − Y j =1 q s j [1 − q − j ] . We may now substitute i := ( n − i −
1) and it suffices to prove that i Y k =1 (cid:2) q n − k +1 − (cid:3) = X S ∈ ( [ n ] i ) i Y j =1 q s j [1 − q − j ] . This can be restated as i Y k =1 q n +1 − q k q k − X S ∈ ( [ n ] i ) q | S | where | S | denotes the sum of the entries in S . We can subdivide the right hand sumdepending on if n ∈ S or not, X S ∈ ( [ n ] i ) q | S | = X S ∈ ( [ n − i ) q | S | + q n X S ∈ ( [ n − i − ) q | S | . By induction over n and i it suffices to show that i Y k =1 q n +1 − q k q k − i Y k =1 q n − q k q k − q n i − Y k =1 q n − q k q k − (cid:3) The case of line graphs follows immediately from [AP18, Prop. 5.18].
Proposition 34.
Let a = (0 , , , . . . , be a line graph. Then ˆG a ( x ; q ) = G a ( x ; q ) . On Lee’s recursion for orientations.
We would also like to prove that theˆG( x ; q ) fulfill Lee’s recursions. However, this is a surprisingly challenging and weare unable to show this at the present time. A proof that Lee’s recursions hold forˆG( x ; q ) would imply that G a ( x ; q ) = ˆG a ( x ; q ) at least for all abelian area sequences a . Computer experiment with n ≤ a ( x ; q ) indeeddo satisfy these recurrences.The class of melting lollipop graphs can be constructed recursively from thecomplete graphs and the line graphs by just using the recursion in Corollary 19.This is in fact done in [HNY18], so we simply sketch a proof of this statement.Recall that a melting lollipop graph a is given by a i = i − i = 1 , . . . , m − m − − k for i = m i = m + 1 , . . . , m + n for some m, n ≥ ≤ k ≤ m −
1. Melting lollipop graphs for various parametersare shown below. A = m =7 ,k =0 ,n =4 m =8 ,k =6 ,n =3 B = m =7 ,k =1 ,n =3 C = m =7 ,k =2 ,n =3 D = m =7 ,k =3 ,n =3 E = m =7 ,k =6 ,n =3 We can use the recursion in Corollary 19 repeatedly to express LLT polynomialsby adding one and removing one shaded box in row m . For example, C can beexpressed in terms of B and D . Similarly, B can be expressed in terms of A and C ,and we get a system of linear equations expressing B , C and D in terms of only A and E .When k = m − E above) the graph Γ a is a disjoint union of a completegraph and a line graph, which is a base case. Furthermore, when k = 0, (as for A LT POLYNOMIALS, ELEMENTARY SYMMETRIC FUNCTIONS AND MELTING LOLLIPOPS17 above) we obtain a melting lollipop graph with the new parameters m = m + 1, k = m − n = n −
1, which are dealt with by induction over n .5. The Hall–Littlewood case
In [HHL05a], the modified Macdonald polynomials ˜H λ ( x ; q, t ) are expressed asa positive sum of certain LLT polynomials. The modified Macdonald polynomialsspecialize to modified Hall–Littlewood polynomials at q = 0, which in turn areclosely related to the transformed Hall–Littlewood polynomials. Definition 35 (See [DLT94, TZ03] for a background) . Let λ be a partition. The transformed Hall–Littlewood polynomials are defined asH µ ( x ; q ) = X λ K λµ ( q )s λ ( x )where K λµ ( q ) are the Kostka–Foulkes polynomials.The H λ are sometimes denoted Q λ and is the adjoint basis to the Hall–Littlewood P polynomials for the standard Hall scalar product, see [DLT94]. A more convenientdefinition of the transformed Hall–Littlewood polynomials is the following. For λ ‘ n we have H λ ( x ; q ) = Y ≤ i Given a partition µ ‘ n , let s i be defined as s i := µ + · · · + µ i ,with s := 1. From µ , we construct a vertical strip diagram Γ µ on n vertices withthe following edges:(a) for each j = 1 , . . . , ‘ ( µ ), let the vertices { s j − , . . . , s j } constitute a completesubgraph of Γ µ ,(b) for each j = 2 , . . . , ‘ ( µ ), we also have the (cid:0) µ j (cid:1) edges { s j − − i → s j − + k + 1 whenever 0 ≤ i, k and i + k ≤ µ j − } . Thus, for each j ≥ µ j = 5. s j →→ →→ → →→ → → → s j − (25)Furthermore, all outer corners are taken as strict edges, see Example 38 below.As before, let O (Γ µ ) denote the set of orientations of the edges of Γ µ . Proposition 37. Let µ be a partition and let Γ µ be the vertical strip diagramconstructed from µ and let G µ ( x ; q ) be the corresponding LLT polynomial. Then ω G µ ( x ; q ) = q P i ≥ ( µi )H µ ( x ; q ) . (26) Brief proof sketch. We use [Hag07, A.59] which states that for any partition λ , thecoefficient of t n ( λ ) in the modified Macdonald polynomial ˜H λ ( x ; q, t ) is almost atransformed Hall–Littlewood polynomial:[ t n ( λ ) ] ˜H λ ( x ; q, t ) = ω H λ ( x ; q ) . The ˜H λ ( x ; q, t ) is a sum over certain LLT polynomials and in particular, the coefficientof t n ( λ ) is a single vertical-strip LLT polynomial, multiplied with q − A , where A isthe sum of arm lengths in the diagram λ . Unraveling the definitions in [Hag07,A.14], we arrive at the identity in (26). (cid:3) Example 38. The Hall–Littlewood polynomial H ( x ; q ) is related to the verticalstrip diagram Γ in (26).Γ = → → · → → · → · · P i ≥ (cid:0) µ i (cid:1) such edges. Notice that the vertex partition of this orientation is { , , , } .Furthermore, it is fairly straightforward to see that for any orientation θ of Γ µ , wemust have that the partition π ( θ ) dominates µ .We can now easily give some strong support for Conjecture 10. Corollary 39. For any partition µ , the vertical-strip LLT polynomial G µ ( x ; q + 1) is e -positive. It was pointed out by the referee that (26) also follows directly from [Hag07, Thm. 6.8]. LT POLYNOMIALS, ELEMENTARY SYMMETRIC FUNCTIONS AND MELTING LOLLIPOPS19 Proof. Using (26), it suffices to prove that H λ ( x ; q + 1) is h-positive. From (23),we have thatH µ ( x ; q + 1) = Y i Find a bijective proof that ˆG µ ( x ; q + 1) is equal to G µ ( x ; q + 1), byinterpreting each term in Equation (31), and combine with (26).It is tempting to believe that summing over the orientations of Γ µ in Definition 36where all edges in condition (b) are oriented in a non-descending manner would giveexactly ω H µ ( x ; q + 1). However, this fails for µ = 222.6. Generalized cocharge and e -positivity In [HNY18], the authors consider a certain classes of unicellular LLT polynomialsthat can be expressed in a particularly nice way. These are polynomials indexed bycomplete graphs, line graphs and a few other families. In this section, we prove thatthe corresponding LLT polynomials are positive in the elementary basis. In fact, wedo this by giving a rather surprising relationship between a type of cocharge andorientations.For a semi-standard Young tableau T , the reading word is formed by reading theboxes of λ row by row from bottom to top, and from left to right within each row.The descent set of a standard Young tableau T is defined asDes( T ) := { i ∈ [ n − 1] : i + 1 appear before i in the reading word } . Given a Dyck diagram a , we define the weight aswt a ( T ) = X i ∈ Des( T ) a n +1 − i . (32)The weight here is also known as cocharge whenever a is the complete graph(0 , , , . . . , n − T denote the transposedtableau, then for any T and a , we haveDes( T ) = [ n − \ Des( T ) and wt a ( T ) = ( a + · · · + a n ) − wt a ( T ) . It will be convenient to define f wt a ( T ) := wt a ( T ) = X i/ ∈ Des( T ) a n +1 − i . (33) Example 41. Let a = (0 , , , , , , , 3) and T = 1 3 42 6 857The reading word of T is 75268134, Des( T ) = { , , } so wt a ( T ) = a + a + a = 7and f wt a ( T ) = 9. Definition 42. Given an area sequence a of length n , we define the polynomial˜G a ( x ; q ) := X λ ‘ n X T ∈ SYT( λ ) q wt a ( T ) s λ ( x ) . (34)From this definition, it follows that ω ˜G a ( x ; q ) = X λ ‘ n X T ∈ SYT( λ ) q e wt a ( T ) s λ ( x ) . (35)The following proposition is a collection of results in [HNY18]. Proposition 43. We have that ˜G a ( x ; q ) = G a ( x ; q ) for the the families of graphslisted in Section 2.1: the complete graphs, line graphs, lollipop graphs, meltingcomplete graphs and melting lollipop graphs. Given a composition γ , let D ( γ ) := { γ , γ + γ , . . . , γ + γ + . . . + γ ‘ } . Lemma 44. Let λ ‘ n and let γ be a composition of n with ‘ parts. Then thestandardization map std : { S ∈ SSYT( λ, γ ) } → { T ∈ SYT( λ ) : Des( T ) ⊆ D ( γ ) } is a bijection.Proof. This is straightforward from the definition of standardization and descents,see for example [Hag07, p. 5]. (cid:3) We shall now introduce a different statistic on orientations. Given θ ∈ O (Γ a ),we say that a vertex v is a bottom of θ if there is no u < v such that ( u, v )is ascending in θ . Let s , . . . , s k be the bottoms ordered decreasingly and let s := n + 1. By definition, vertex 1 is always a bottom. Let σ ( θ ) be defined as thecomposition of n with the parts given by { s i − − s i : i = 1 , . . . , k } and note that D ( σ ( θ )) = { n + 1 − s i : i = 1 , , . . . , k − } . Example 45. The orientation θ in (36) has vertices 1, 3 and 6 as bottoms. Fur-thermore, σ ( θ ) = (1 , , 2) and D ( σ ( θ )) = { , } .6 →→ →→ → 21 (36)Note that π ( θ ) = (5 , 1) so σ and π are indeed very different. LT POLYNOMIALS, ELEMENTARY SYMMETRIC FUNCTIONS AND MELTING LOLLIPOPS21 The following theorem was proved for the complete graph and the line graph in[AP18]. We can now generalize it to all unit interval graphs. Theorem 46. Let a be an area sequence of length n . Then ˜G a ( x ; q + 1) = X θ ∈ O (Γ a ) q asc( θ ) e σ ( θ ) ( x ) . (37) Proof. We apply ω on both sides of Equation (37), so it suffices to prove that ω ˜G a ( x ; q + 1) = X θ ∈ O (Γ a ) q asc( θ ) h σ ( θ ) ( x ) . (38)Recall, in e.g. [Mac95], the standard expansionh ν ( x ) = X λ K λ,ν s λ ( x ) , (39)where K λ,ν = | SSYT( λ, ν ) | are the Kostka coefficients. Thus, comparing both sidesof (38) in the Schur basis, it suffices to show that for every partition λ , X T ∈ SYT( λ ) (1 + q ) e wt a ( T ) = X θ ∈ O (Γ a ) q asc( θ ) K λ,σ ( θ ) . Using Lemma 44 in the right hand side and unraveling the definition in the lefthand side, it is enough to prove that X T ∈ SYT( λ ) Y i/ ∈ Des( T ) (1 + q ) a n +1 − i = X T ∈ SYT( λ ) X θ ∈ O (Γ a )Des( T ) ⊆ D ( σ ( θ )) q asc( θ ) . It then suffices to prove that for a fixed T ∈ SYT( λ ) we have Y i/ ∈ Des( T ) (1 + q ) a n +1 − i = X θ ∈ O (Γ a )Des( T ) ⊆ D ( σ ( θ )) q asc( θ ) . (40)Both sides may now be interpreted as a weighted sum over all orientations of Γ a where no ascending edges end in { i : n + 1 − i ∈ Des( T ) } . (cid:3) Corollary 47. All families of unicellular LLT polynomials G a ( x ; q + 1) indexedby complete graphs, line graphs, lollipop graphs and melting lollipop graphs are e -positive. Notice that the formula in (37) is different from the conjectured formula inConjecture 10, since π ( θ ) and σ ( θ ) are different. This is not surprising as G a ( x ; q )and ˜G a ( x ; q ) are not equal for general a . However, it is rather remarkable thatConjecture 10 implies that (37) and Equation (6) agree whenever G a ( x ; q ) =˜G a ( x ; q ). 7. A possible approach to settle the main conjecture In [AP18] and later in [AS19] (with a different approach) we gave formulas forthe power-sum expansion of all vertical-strip LLT polynomials. The unicellularcase is a straightforward consequence of Lemma 12 (see [AP18, HW17]) togetherwith the power-sum expansion formula for the chromatic symmetric symmetricfunctions. We note that the formula in the chromatic case was first conjectured byShareshian–Wachs and later proved by Athanasiadis [Ath15]. It is straightforward to expand (6) in the power-sum basis, so to settle Conjec-ture 10, it suffices to show that ω G a ( x ; q + 1) = ω ˆG a ( x ; q + 1) for all a by comparingcoefficients of p λ /z λ . We shall now introduce the necessary terminology from [AS19]to state Conjecture 10 in this form.For any subset S ⊆ E (Γ a ), let P ( S ) denote the poset given by the transitiveclosure of the edges in S . Given a poset P on n elements, let O ( P ) be the set oforder-preserving surjections f : P → [ k ] for some k . The type of a surjection f isdefined as α ( f ) := ( | f − (1) | , | f − (2) | , . . . , | f − ( k ) | ) , and this is a composition of n with k parts. Let O α ( P ) ⊆ O ( P ) be the setof surjections of type α . Finally, let O ∗ α ( P ) ⊆ O α ( P ) be the set of surjections f ∈ O α ( P ) such that for each j ∈ [ k ], f − ( j ) is a subposet of P with a uniqueminimal element. Proposition 48 (See [AS19, Thm. 5.6, Thm. 7.10]) . The power-sum expansion of ω G a ( x ; q + 1) is given as ω G a ( x ; q + 1) = X θ ∈ O ( a ) q asc( θ ) X λ ‘ n p λ ( x ) z λ |O ∗ λ ( P ( θ )) | (41) where P ( θ ) is the poset on [ n ] and edges given by the transitive closure of theascending edges in θ . The family of functions ˆG a ( x ; q + 1) has a similar expansion in terms of thepower-sum symmetric functions. Lemma 49. The power-sum expansion of ω ˆG a ( x ; q + 1) is given as ω ˆG a ( x ; q + 1) = X θ ∈ O ( a ) q asc( θ ) X λ ‘ n p λ ( x ) z λ |O ∗ λ ( B ( θ )) | (42) where B ( θ ) is the poset consisting of a disjoint union of chains with lengths givenby π ( θ ) .Proof. This follows easily from the definition of ˆG a ( x ; q + 1) and the expansionof the elementary symmetric functions into power-sum symmetric functions, see[ER91] and [AS19, Section 7]. (cid:3) Conjecture 50 (Equivalent with Conjecture 10) . 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