aa r X i v : . [ m a t h . C O ] S e p LECTURE HALL P -PARTITIONS PETTER BR ¨AND´EN AND MADELEINE LEANDER
Abstract.
We introduce and study s -lecture hall P -partitions which is a gen-eralization of s -lecture hall partitions to labeled (weighted) posets. We providegenerating function identities for s -lecture hall P -partitions that generalizeidentities obtained by Savage and Schuster for s -lecture hall partitions, and byStanley for P -partitions. We also prove that the corresponding ( P, s )-Eulerianpolynomials are real-rooted for certain pairs (
P, s ), and speculate on unimodal-ity properties of these polynomials. Introduction
Let s = ( s , . . . , s n ) be a sequence of positive integers. An s - lecture hall partition is an integer sequence λ = ( λ , . . . , λ n ) satisfying 0 ≤ λ /s ≤ · · · ≤ λ n /s n . Theseare generalizations of lecture hall partitions , corresponding to the case when s =(1 , , . . . , n ), first studied by Bousquet-M´elou and Eriksson [3]. It has recentlybeen made evident that s -lecture hall partitions serve as a rich model for variouscombinatorial structures with nice generating functions, see [2, 3, 4, 13, 18, 17, 19,20] and the references therein.In this paper we generalize the concept of s -lecture hall partitions to labeledposets. This constitutes a generalization of Stanley’s theory of P -partitions, see[22, Ch. 3.15]. In Section 3 we derive multivariate generating function identi-ties for s -lecture hall P -partitions, and prove a reciprocity theorem (Theorem 3.9).When P is a naturally labeled chain or an anti-chain, the generating function iden-tities obtained produce results on s -lecture hall partitions and signed permutations,respectively (see Section 6). We also introduce and study a ( P, s )-Eulerian poly-nomial. In Section 4 we prove that this polynomial is palindromic for sign-gradedlabeled posets with a specific choice of s . In Section 5 we prove that the ( P, s )-Eulerian polynomial is real-rooted for certain choices of (
P, s ), and we also speculateon unimodality properties satisfied by these polynomials.2.
Lecture hall P -partitions In this paper a labeled poset is a partially ordered set on [ p ] := { , . . . , p } forsome positive integer p , i.e., P = ([ p ] , (cid:22) ), where (cid:22) denotes the partial order. Wewill use the symbol ≤ to denote the usual total order on the integers. If P is alabeled poset, then a P -partition is a map f : [ p ] → R such that(1) if x ≺ y , then f ( x ) ≤ f ( y ), and(2) if x ≺ y and x > y , then f ( x ) < f ( y ). What we call P -partitions are called reverse ( P, ω )-partitions in [22, 23]. However the theoryof (
P, ω )-partitions and reverse (
P, ω )-partitions are clearly equivalent.
PETTER BR¨AND´EN AND MADELEINE LEANDER
The theory of P -partitions was developed by Stanley in his thesis and has sincethen been used frequently in several different combinatorial settings, see [22, 23].Let O ( P ) = { f ∈ R p : f is a P -partition and 0 ≤ f ( x ) ≤ x ∈ [ p ] } be the order polytope associated to P . Note that if P is naturally labeled, i.e., x ≺ y implies x < y , then O ( P ) is a closed integral polytope. Otherwise O ( P )is the intersection of a finite number of open or closed half-spaces. Recall thatthe Ehrhart polynomial of an integral polytope P in R p is defined for nonnegativeintegers n as i ( P , n ) = | n P ∩ Z p | , where n P = { n x : x ∈ P} , see [22, p. 497]. For order polytopes we have thefollowing relationship due to Stanley: X n ≥ i ( O ( P ) , n ) t n = A P ( t )(1 − t ) p +1 , where A P ( t ) is the P -Eulerian polynomial, which is the generating polynomial ofthe descent statistic over the set of all linear extensions of P , see [22, Ch. 3.15].The purpose of this paper is to initiate the study of a lecture hall generalizationof P -partitions. Let P be a labeled poset and let s : [ p ] → Z + := { , , , . . . } bean arbitrary map. We define a lecture hall ( P, s )- partition to be a map f : [ p ] → R such that(1) if x ≺ y , then f ( x ) /s ( x ) ≤ f ( y ) /s ( y ), and(2) if x ≺ y and x > y , then f ( x ) /s ( x ) < f ( y ) /s ( y ).Let O ( P, s ) = { f ∈ R p : f is a ( P, s )-partition and 0 ≤ f ( x ) /s ( x ) ≤ x ∈ [ p ] } be the lecture hall order polytope associated to ( P, s ). We also let C ( P, s ) = { f ∈ R p : f is a ( P, s )-partition and 0 ≤ f ( x ) /s ( x ) for all x ∈ [ p ] } be the lecture hall order cone associated to ( P, s ). The (
P, s )- Eulerian polynomial , A ( P,s ) ( t ), is defined by X n ≥ i ( O ( P, s ) , n ) t n = A ( P,s ) ( t )(1 − t ) p +1 . The main generating functions
In this section we derive formulas for the main generating functions associated tolecture hall (
P, s )-partitions. The outline follows Stanley’s theory of P -partitions[22, Ch. 3.15]. We shall see in Section 6 that the special cases when P is natu-rally labeled chain or an anti-chain automatically produce results on lecture hallpolytopes and signed permutations, respectively.Let S p denote the symmetric group on [ p ]. If π = π π · · · π p ∈ S p is a permu-tation written in one-line notation, we let P π denote the labeled chain π ≺ π ≺· · · ≺ π p . If P = ([ p ] , (cid:22) ) is a labeled poset, let L ( P ) denote the set L ( P ) := { π ∈ S p : if π i (cid:22) π j , then i ≤ j, for all i, j ∈ [ p ] } , of linear extensions (or the Jordan-H¨older set ) of P . The following lemma is an im-mediate consequence of Stanley’s decomposition of P -partitions [22, Lemma 3.15.3]. Lemma 3.1. If P is a labeled poset and s : [ p ] → Z + , then C ( P, s ) = G π ∈L ( P ) C ( P π , s ) , where F denotes disjoint union. Let s : [ p ] → Z + . An s - colored permutation is a pair τ = ( π, r ) where π ∈ S p ,and r : [ p ] → N satisfies r ( π i ) ∈ { , , . . . , s ( π i ) − } for all 1 ≤ i ≤ p . If P = ([ p ] , (cid:22) )is a labeled poset, let L ( P, s ) = { τ : τ = ( π, r ) where π ∈ L ( P ) and τ is an s -colored permutation } . For f : [ p ] → N , let q ( f ) , r ( f ) : [ p ] → N be the unique functions satisfying f ( x ) = q ( f )( x ) · s ( x ) + r ( f )( x ) , where q ( f )( x ) ∈ N and 0 ≤ r ( f )( x ) < s ( x ) , for all x ∈ [ p ]. Let further F ( P,s ) ( x , y ) = X f ∈ N ( P,s ) y r ( f ) x q ( f ) , where x r = x r (1)1 x r (2)2 · · · x r ( p ) p and N ( P, s ) = C ( P, s ) ∩ N p . We say that i ∈ [ p − descent of τ = ( π, r ) if ( π i < π i +1 and r ( π i ) /s ( π i ) > r ( π i +1 ) /s ( π i +1 ) , or, π i > π i +1 and r ( π i ) /s ( π i ) ≥ r ( π i +1 ) /s ( π i +1 ) , Let D ( τ ) = { i ∈ [ p −
1] : i is a descent } . Theorem 3.2. If P is a labeled poset and s : [ p ] → Z + , then F ( P,s ) ( x , y ) = X τ =( π,r ) ∈L ( P,s ) y r Y i ∈ D ( τ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p ) . (1) Proof.
By Lemma 3.1 we may assume that P = P π is a labeled chain. Let f ∈ N p ,and write f ( t ) = q ( t ) s ( t ) + r ( t ), where 0 ≤ r ( t ) < s ( t ) and q ( t ) ∈ N for all t ∈ [ p ].What conditions on q and r guarantee f ∈ N ( P, s )? Suppose π i < π i +1 . Then weneed q ( π i ) + r ( π i ) s ( π i ) = f ( π i ) s ( π i ) ≤ f ( π i +1 ) s ( π i +1 ) = q ( π i +1 ) + r ( π i +1 ) s ( π i +1 ) . (2)If r ( π i ) /s ( π i ) ≤ r ( π i +1 ) /s ( π i +1 ), then (2) holds if and only if q ( π i ) ≤ q ( π i +1 ). If r ( π i ) /s ( π i ) > r ( π i +1 ) /s ( π i +1 ), then (2) holds if and only if q ( π i ) < q ( π i +1 ).Suppose π i > π i +1 . Then we need q ( π i ) + r ( π i ) s ( π i ) = f ( π i ) s ( π i ) < f ( π i +1 ) s ( π i +1 ) = q ( π i +1 ) + r ( π i +1 ) s ( π i +1 ) . (3)If r ( π i ) /s ( π i ) < r ( π i +1 ) /s ( π i +1 ), then (3) holds if and only if q ( π i ) ≤ q ( π i +1 ). If r ( π i ) /s ( π i ) ≥ r ( π i +1 ) /s ( π i +1 ), then (3) holds if and only if q ( π i ) < q ( π i +1 ).Let τ = ( π, r ), where r is fixed. Then f = qs + r ∈ N ( P, s ) with given (fixed) r if and only if 0 ≤ q ( π ) ≤ q ( π ) ≤ · · · ≤ q ( π p ) , (4) PETTER BR¨AND´EN AND MADELEINE LEANDER where q ( π i ) < q ( π i +1 ) if i ∈ D ( τ ). Hence f = qs + r ∈ N ( P, s ) if and only if foreach k ∈ [ p ]: q ( π k ) = α k + |{ i ∈ D ( τ ) : i < k }| , where α k ∈ N and 0 ≤ α ≤ · · · ≤ α p . Hence X q p Y i =1 x q ( π i ) π i = X ≤ α ≤···≤ α p x α π · · · x α p π p Y i ∈ D ( τ ) x π i +1 · · · x π p = Y i ∈ D ( τ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p ) , where the first sum is over all q satisfying (4). The theorem follows. (cid:3) Let Z + ( P, s ) = C ( P, s ) ∩ Z p + and let F +( P,s ) ( x , y ) = X f ∈ Z + ( P,s ) y r ( f ) x q ( f ) . Let further D ( τ ) = ( D ( τ ) , if r ( π ) = 0 ,D ( τ ) ∪ { } , if r ( π ) = 0 . Theorem 3.3. If P is a labeled poset and s : [ p ] → Z + , then F +( P,s ) ( x , y ) = X τ =( π,r ) ∈L ( P,s ) y r Y i ∈ D ( τ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p ) . Proof.
Consider ( P ′ , s ′ ) where P ′ is obtained from P by adjoining a least element ˆ0labeled p + 1, and s ′ : [ p + 1] → Z + is such that s ′ restricted to [ p ] agrees with s . Letalso s ′ ( p + 1) > max { s ( t ) : t ∈ [ p ] } . Then f ∈ N ( P ′ , s ′ ) if and only if f | [ p ] ∈ N ( P, s )and 0 ≤ f ( p + 1) s ′ ( p + 1) < f ( x ) s ( x ) , for all x ∈ [ p ] . Thus F +( P,s ) ( x , y ) is obtained from F ( P ′ ,s ′ ) ( x , y ) when we restrict to all f ∈ N ( P ′ , s ′ )with f ( p + 1) = 1, i.e., q ( p + 1) = 0 and r ( p + 1) = 1, and then shift the indices.Hence i = 0 is a descent in (( p + 1) π π · · · π p , r ) if and only if r ( π ) = 0, and theproof follows. (cid:3) For f : [ p ] → Z + , let q ′ ( f ) , r ′ ( f ) : [ p ] → N be the unique functions satisfying f ( x ) = q ′ ( f )( x ) · s ( x ) + r ′ ( f )( x ) , where q ′ ( f )( x ) ∈ N and 0 < r ′ ( f )( x ) ≤ s ( x ) , for all x ∈ [ p ]. Let further G ( P,s ) ( x , y ) = X f ∈ Z + ( P,s ) y r ′ ( f ) x q ′ ( f ) . Let D ( τ ) be the set of all i ∈ [ p −
1] for which π i < π i +1 and ( r ( π i ) + 1) /s ( π i ) > ( r ( π i +1 ) + 1) /s ( π i +1 ) , or, π i > π i +1 and ( r ( π i ) + 1) /s ( π i ) ≥ ( r ( π i +1 ) + 1) /s ( π i +1 ) . Theorem 3.4. If P is a labeled poset and s : [ p ] → Z + , then G ( P,s ) ( x , y ) = X τ =( π,r ) ∈L ( P,s ) y r + Y i ∈ D ( τ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p ) , where = (1 , , . . . , is the all ones vector.Proof. The proof is almost identical to that of Theorem 3.2, and is therefore omit-ted. (cid:3)
For n ∈ N , let N ≤ n ( P, s ) = { f ∈ N ( P, s ) : f ( x ) /s ( x ) ≤ n for all x ∈ [ p ] } , and let F ( P,s ) ( x , y ; n ) = X f ∈ N ≤ n ( P,s ) y r ( f ) x q ( f ) . The polynomials F +( P,s ) ( x , y ; n ) and G ( P,s ) ( x , y ; n ) are defined analogously over { f ∈ Z + ( P, s ) : f ( x ) /s ( x ) ≤ n for all x ∈ [ p ] } . Let also N
Proposition 3.5. If P is a labeled poset and s : [ p ] → Z + , then X n ≥ F ( P,s ) ( x , y ; n ) t n = X τ =( π,r ) ∈L ( P,s ) y r Y i ∈ D ( τ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p t ) t | D ( τ ) | − t , (5) X n ≥ F ′ ( P,s ) ( x , y ; n ) t n = X τ =( π,r ) ∈L ( P,s ) y r Y i ∈ D ( τ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p t ) t | D ( τ ) | +1 − t , (6) X n ≥ F +( P,s ) ( x , y ; n ) t n = X τ =( π,r ) ∈L ( P,s ) y r Y i ∈ D ( τ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p t ) t | D ( τ ) | − t , (7) X n ≥ G ( P,s ) ( x , y ; n ) t n = X τ =( π,r ) ∈L ( P,s ) y r + Y i ∈ D ( τ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p t ) t | D ( τ ) | +1 − t . (8) Proof.
For (5) consider ( P ′ , s ′ ) where P ′ is obtained from P by adjoining a greatestelement ˆ1 labeled p + 1, and s ′ : [ p + 1] → Z + restricted to [ p ] agrees with s , while s ′ ( p + 1) = 1. If we set x p +1 = t , then X n ≥ F ( P,s ) ( x , y ; n ) t n = F ( P ′ ,s ′ ) , and L ( P ′ , s ′ ) = { ( π · · · π p ( p + 1) , r ′ ) : ( π · · · π p , r ′ | P ) ∈ L ( P, s ) and r ′ ( p + 1) = 0 } . The identity (5) follows by noting that i = p is a descent of ( π · · · π p ( p + 1) , r ′ ) ifand only if r ( π p ) /s ( π p ) > r ′ ( p + 1) /s ′ ( p + 1) = 0.The other identities follows similarly. For example (6) follows by considering( P ′ , s ′ ) where P ′ is obtained from P by adjoining a greatest element ˆ1 labeled 0(and then relabel so that P ′ has ground set [ p + 1]). For (8) consider again ( P ′ , s ′ ),where P ′ is obtained from P by adjoining a greatest element ˆ1 labeled p +1, and s ′ isdefined as for the case of (5). Note that since r ′ ( p + 1) = 1 we have q ′ ( p + 1) = n − f ( p + 1) = n . This explains the shift by one in the exponent on the right handside of (8), i.e., | D ( τ ) | + 1. (cid:3) If q is a variable, let [0] q := 0 and [ n ] q := 1 + q + q + · · · + q n − for n ≥
1. Forthe special case of (5) when P is an anti-chain we acquire the following corollary,which is a generalization of [1, Theorem 5.23]. Corollary 3.6. If P is an anti-chain and s : [ p ] → Z + , then X n ≥ p Y i =1 ( x ni + [ n ] x i [ s ( i )] y i ) t n = X τ =( π,r ) ∈L ( P,s ) y r Y i ∈ D ( τ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p t ) t | D ( τ ) | − t Proof.
Let P be an anti-chain and let s : [ p ] → Z + . Consider f ∈ N ≤ n ( P, s ). Since P is an anti-chain, f ( i ) and f ( j ) are independent for all 1 ≤ i < j ≤ p , and theonly restriction is 0 ≤ f ( i ) ≤ ns ( i ) for all 1 ≤ i ≤ p . We write f ( i ) = s ( i ) q ( i ) + r ( i ),where 0 ≤ r ( i ) < s ( i ). Then f ∈ N ≤ n ( P, s ) if and only if either q ( i ) = n and r ( i ) = 0, or 0 ≤ q ( i ) ≤ n − ≤ r ( i ) ≤ s ( i ) −
1. Hence X f ∈ N ≤ n ( P,s ) y r ( f ) x q ( f ) = p Y i =1 (cid:0) x i [ s ( i )] y i + · · · + x n − i [ s ( i )] y i + x ni (cid:1) = p Y i =1 ( x ni + [ n ] x i [ s ( i )] y i ) . The corollary now follows from (5). (cid:3)
Note that the special case of (5) when P is a naturally labeled chain gives ananalogue (by an appropriate change of variables) to one of the main results in [19],see Theorem 5 therein. From (5) we also get an interpretation of the Eulerianpolynomial A ( P,s ) ( t ). For τ ∈ L ( P, s ), let des s ( τ ) = | D ( τ ) | . Corollary 3.7. If P is a labeled poset and s : [ p ] → Z + , then A ( P,s ) ( t ) = X τ ∈L ( P,s ) t des s ( τ ) . The next corollary follows from Proposition 3.5 by setting the x - and y -variablesto 1. Corollary 3.8. If P is a labeled poset and s : [ p ] → Z + , then X τ ∈L ( P,s ) t | D ( τ ) | = X τ ∈L ( P,s ) t | D ( τ ) | +1 , and if s ( x ) = 1 for all minimal elements x in P , then A ( P,s ) ( t ) = X τ ∈L ( P,s ) t | D ( τ ) | = X τ ∈L ( P,s ) t | D ( τ ) | . (9)Let P = ([ p ] , (cid:22) ) be a labeled poset. For i ∈ [ p ], let i ∗ = p + 1 − i , and let ( P ∗ , s ∗ )be defined by P ∗ = ([ p ] , (cid:22) ∗ ) with i (cid:22) j in P if and only if i ∗ (cid:22) ∗ j ∗ in P ∗ , for all i, j ∈ [ p ] , and s ∗ ( i ∗ ) = s ( i ) for all i ∈ [ p ]. The poset P ∗ is called the dual of P . Theorem 3.9 (Reciprocity theorem) . If P is a labeled poset and s : [ p ] → Z + ,then G ( P ∗ ,s ∗ ) ( x ∗ , y ∗ ) = ( − p y s (1)1 · · · y s ( p ) p x · · · x p F ( P,s ) ( x − , y − ) , where x ∗ = ( x p , x p − , . . . , x ) and x − = ( x − , . . . , x − p ) .Proof. For τ = ( π, r ) ∈ L ( P, s ), let τ ∗ = ( π ∗ π ∗ · · · π ∗ p , r ∗ ) where r ∗ ( i ∗ ) = s ( i ) − − r ( i ) for all i ∈ [ p ]. Clearly the map τ τ ∗ is a bijection between L ( P, s ) and L ( P ∗ , s ∗ ). Moreover if i ∈ [ p − i ∈ D ( τ ) if and only if ( π i < π i +1 and ( r ( π i ) + 1) /s ( π i ) > ( r ( π i +1 ) + 1) /s ( π i +1 ) , or, π i > π i +1 and ( r ( π i ) + 1) /s ( π i ) ≥ ( r ( π i +1 ) + 1) /s ( π i +1 ) , PETTER BR¨AND´EN AND MADELEINE LEANDER if and only if ( π ∗ i > π ∗ i +1 and r ∗ ( π ∗ i ) /s ∗ ( π ∗ i ) < r ∗ ( π ∗ i +1 ) /s ∗ ( π ∗ i +1 ) , or, π ∗ i < π ∗ i +1 and r ∗ ( π ∗ i ) /s ∗ ( π ∗ i ) ≤ r ∗ ( π ∗ i +1 ) /s ∗ ( π ∗ i +1 )if and only if i ∈ [ p − \ D ( τ ∗ ). Thus D ( τ ) = [ p − \ D ( τ ∗ ) and D ( τ ) = [ p − \ D ( τ ∗ ) , (10)for all τ ∈ L ( P, s ). Now F ( P,s ) ( x , y ) = X τ ∈L ( P,s ) y r Y i ∈ D ( τ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p ) = X τ ∈L ( P,s ) y r Y i ∈ [ p − \ D ( τ ∗ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p )= X τ ∈L ( P,s ) y s ( y ∗ ) − ( r ∗ + ) x · · · x p Y i ∈ D ( τ ∗ ) x − π i +1 · · · x − π p Y i ∈ [ p ] (1 − x π i · · · x π p ) Y i ∈ [ p ] x π i · · · x π p = ( − p y s (1)1 · · · y s ( p ) p x · · · x p X τ ∈L ( P,s ) ( y ∗ ) − ( r ∗ + ) Y i ∈ D ( τ ∗ ) x − π i +1 · · · x − π p Y i ∈ [ p ] (1 − x − π i · · · x − π p )= ( − p y s (1)1 · · · y s ( p ) p x · · · x p G ( P ∗ ,s ∗ ) (( x ∗ ) − , ( y ∗ ) − ) , from which the theorem follows. (cid:3) Remark . Theorem 3.9 generalizes the reciprocity theorem in [4] which followsas the special case when P is a naturally labeled chain.4. Sign-ranked posets
Let P = { ≺ ≺ · · · ≺ p } be a naturally labeled chain, and let s ( i ) = i for all i ∈ [ p ]. Savage and Schuster [19, Lemma 1] proved that A ( P,s ) ( t ) is equal to theEulerian polynomial A p ( t ) = X π ∈ S p t des( π ) , where des( π ) = |{ i ∈ [ p ] : π i > π i +1 } . Recall that a polynomial g ( t ) is palindromic if t N g (1 /t ) = g ( t ) for some integer N . It is well known that A p ( t ) is palindromic(in fact t p − A p (1 /t ) = A p ( t )). The same is known to be true for the P -Eulerianpolynomial of any naturally labeled graded poset, see [22, Corollary 3.15.18], andmore generally for P -Eulerian polynomials of so called sign-graded labeled posets[10, Corollary 2.4]. We shall here generalize these results to ( P, s )-Eulerian poly-nomials.Recall that a pair of elements elements ( x, y ) taken from a labeled poset P is a covering relation if x ≺ y and x ≺ z ≺ y for no z ∈ P . Let E ( P ) denote the set of covering relations of P . If P is a labeled poset define a function ǫ : E ( P ) → {− , } by ǫ ( x, y ) = ( , if x < y, and − , if x > y. Sign-graded (labeled) posets, introduced in [10], generalize graded naturally labeledposets. A labeled poset P is sign-graded of rank r , if k X i =1 ǫ ( x i − , x i ) = r for each maximal chain x ≺ x ≺ · · · ≺ x k in P . A sign-graded poset is equippedwith a well-defined rank-function , ρ : P → Z , defined by ρ ( x ) = k X i =1 ǫ ( x i − , x i ) , where x ≺ x ≺ · · · ≺ x k = x is any unrefinable chain, x is a minimal element and x k = x . Hence a naturally labeled poset is sign-graded if and only if it is graded.A labeled poset P is sign-ranked if for each maximal element x ∈ P , the subposet { y ∈ P : y (cid:22) x } is sign-graded. Note that each sign-ranked poset has a well-definedrank function ρ : P → Z . Thus a naturally labeled poset is sign-ranked if and onlyif it is ranked. Theorem 4.1.
Let P be a sign-ranked labeled poset and suppose its rank functionattains non-negative values only. Let s ( x ) = ρ ( x ) + 1 for each x ∈ [ p ] , and define u : N ( P, s ) → Z p by u ( f )( x ∗ ) = f ( x ) + ρ ( x ) . Then u : N ≤ n ( P, s ) → N 1, then f ( x ) /s ( x ) < f ( y ) /s ( y ).Hence it suffices to consider covering relations when proving that u : N ( P, s ) → N ( P ∗ , s ∗ ).Let f ∈ N ( P, s ). Suppose y covers x and ǫ ( x, y ) = 1. Then f ( x ) /s ( x ) ≤ f ( y ) /s ( y ) and s ( x ) < s ( y ), and thus u ( f )( x ∗ ) s ∗ ( x ∗ ) = f ( x ) + s ( x ) − s ( x ) ≤ f ( y ) s ( y ) + 1 − s ( x ) < f ( y ) s ( y ) + 1 − s ( y ) = u ( f )( y ∗ ) s ∗ ( y ∗ ) , as desired.Suppose y covers x and ǫ ( x, y ) = − 1. Then f ( x ) /s ( x ) < f ( y ) /s ( y ) and s ( x ) = s ( y ) + 1 so that u ( f )( y ∗ ) s ∗ ( y ∗ ) − u ( f )( x ∗ ) s ∗ ( x ∗ ) = f ( y ) s ( y ) − f ( x ) s ( y ) + 1 − (cid:18) s ( y ) − s ( y ) + 1 (cid:19) . We want to prove that the quantity on either side of the equality above is nonneg-ative. By assumption f ( y ) s ( y ) − f ( x ) s ( y ) + 1 = ( s ( y ) + 1) f ( y ) − s ( y ) f ( x ) s ( y )( s ( y ) + 1) > . Hence ( s ( y ) + 1) f ( y ) − s ( y ) f ( x ) is a positive integer, so that f ( y ) s ( y ) − f ( x ) s ( y ) + 1 ≥ s ( y )( s ( y ) + 1) , as desired. Note that u ( f ) is nonnegative since it is increasing and u ( f )( x ∗ ) = f ( x )when x ∗ is a minimal element in P ∗ . Hence u ( f ) ∈ N ( P ∗ , s ∗ ).Let η : N ( P ∗ , s ∗ ) → Z P be defined by η ( g )( x ) = g ( x ∗ ) − ρ ( x ) = g ( x ∗ ) + ρ ∗ ( x ∗ ),where ρ ∗ is the rank function of P ∗ . Clearly η : N ( P ∗ , s ∗ ) → N ( P, s ) by the exactsame arguments as above. Thus u − = η and u : N ( P, s ) → N ( P ∗ , s ∗ ) is a bijection.Now u ( f )( x ∗ ) /s ∗ ( x ∗ ) = f ( x ) /s ( x ) + ( s ( x ) − /s ( x ) < n + 1 if f ∈ N ≤ n ( P, s )and x ∈ P , so that u : N ≤ n ( P, s ) → N Theorem 4.2. If P is a sign-ranked labeled poset with nonnegative rank function ρ and s = ρ + 1 , then A ( P,s ) ( t ) = t p − A ( P,s ) ( t − ) and ( − p i ( O ( P, s ) , − t ) = i ( O ( P, s ) , t − . Proof. By (5), (6) and Theorem 4.1 A ( P,s ) ( t ) = X τ ∈L ( P,s ) t | D ( τ ) | = X τ ∗ ∈L ( P ∗ ,s ∗ ) t | D ( τ ∗ ) | . The first part of the theorem now follows from (9) and (10). The second partfollows from e.g., [22, Lemma 3.15.11]. (cid:3) Real-rootedness and unimodality The Neggers-Stanley conjecture asserted that for each labeled poset P , the Euler-ian polynomial A P ( t ) is real-rooted. Although the conjecture is refuted in its fullgenerality [9, 24], it is known to hold for certain classes of posets [6, 25]. Moreover,when P is sign-graded, then the coefficients of A P ( t ) form a unimodal sequence[10, 15]. It is natural to ask for which pairs ( P, s )(a) is A ( P,s ) ( t ) real-rooted?(b) do the coefficients of A ( P,s ) ( t ) form a unimodal sequence?We first address (a). Suppose P = ([ p ] , (cid:22) P ), Q = ([ q ] , (cid:22) Q ) and R = ([ p + q ] , (cid:22) R )are labeled posets such that [ p + q ] is the disjoint union of the two sets { u < u < · · · < u p } and { v < v < · · · < v q } , and x (cid:22) R y if and only if either • x = u i and y = u j for some i, j ∈ [ p ] with i (cid:22) P j , or • x = v i and y = v j for some i, j ∈ [ q ] with i (cid:22) Q j .We say that R is a disjoint union of P and Q and write R = P ⊔ Q . Moreover if s P : [ p ] → Z + and s Q : [ q ] → Z + , then we define s P ⊔ Q : [ p + q ] → Z + as the uniquefunction satisfying s P ⊔ Q ( u i ) = s P ( i ) and s P ⊔ Q ( v j ) = s Q ( j ). Proposition 5.1. If the polynomials A ( P,s P ) ( t ) and A ( Q,s Q ) ( t ) are real-rooted, thenso is the polynomial A ( P ⊔ Q,s P ⊔ s Q ) ( t ) . Proof. Clearly i (( P ⊔ Q, s P ⊔ s Q ) , t ) = i ( O ( P, s P ) , t ) · i ( O ( Q, s Q ) , t ) , so the proposition follows from [26, Theorem 0.1]. (cid:3) It was proved in [21] that if P = { ≺ ≺ · · · ≺ p } and s : [ p ] → Z + is arbitrary,then A ( P,s ) ( t ) is real-rooted. In Theorem 5.2 below we generalize this result toordinal sums of anti-chains. If P = ( X, (cid:22) P ) and Q = ( Y, (cid:22) Q ) are posets on disjointground sets, then the ordinal sum , P ⊕ Q = ( X ∪ Y, (cid:22) ), is the poset with relations(1) x ≺ x , for all x , x ∈ X with x ≺ P x ,(2) y ≺ y , for all y , y ∈ X with y ≺ Q y , and(3) x ≺ y for all x ∈ X and y ∈ Y .Let f and g be two real-rooted polynomials in R [ t ] with positive leading coeffi-cients. Let further α ≥ α ≥ · · · ≥ α n and β ≥ β ≥ · · · ≥ β m be the zeros of f and g , respectively. If · · · ≤ α ≤ β ≤ α ≤ β we say that f is an interleaver of g and we write f ≪ g . We also let f ≪ ≪ f . We call a sequence F n = ( f i ) ni =1 of real-rooted polynomials interlacing if f i ≪ f j for all 1 ≤ i < j ≤ n . We denote by F n the family of all interlacingsequences ( f i ) ni =1 of polynomials and we let F + n be the family of ( f i ) ni =1 ∈ F n suchthat f i has nonnegative coefficients for all 1 ≤ i ≤ n .To avoid unnecessary technicalities we here redefine a labeled poset to be a poset P = ( S, (cid:22) ), where S is any set of positive integers. Thus L ( P ) is now the set ofrearrangements of S that are also linear extensions of P .Equip X ( P, s ) := { ( k, x ) : x ∈ P and 0 ≤ k < s ( x ) } with a total order defined by( k, x ) < ( ℓ, y ) if k/s ( x ) < ℓ/s ( y ), or k/s ( x ) = ℓ/s ( y ) and x < y . For γ ∈ X ( P, s ),let A γ ( P,s ) ( t ) = X τ =( π,r ) ∈L ( P,s )( r ( π ) ,π )= γ t des s ( τ ) . Theorem 5.2. Suppose P = A p ⊕ · · · ⊕ A p m is an ordinal sum of anti-chains,and let s : P → Z + be a function which is constant on A p i for ≤ i ≤ m . Then { A γ ( P,s ) ( t ) } γ ∈ X , where X = X ( P, s ) , is an interlacing sequence of polynomials.In particular A ( P,s ) ( t ) and A γ ( P,s ) ( t ) are real-rooted for all γ ∈ X .Proof. The proof is by induction over m . Suppose m = 1, p = n , A n is the anti-chain on [ n ], and s ( A n ) = { s } . We prove the case m = 1 by induction over n . If n = 1 we get the sequence 1 , t, t, . . . , t which is interlacing. Otherwise if γ = ( k, π ),then A γ ( A n ,s ) ( t ) = X κ<γ tA κ ( A n − ,s ′ ) ( t ) + X κ ≥ γ A κ ( A n − ,s ′ ) ( t ) , where s ′ is s restricted to A n − . This recursion preserves the interlacing property,see [21, Theorem 2.3] and [11], which proves the case m = 1 by induction.Suppose m > 1. The proof for m is again by induction over p = n . If p = 1,then A γ ( P,s ) ( t ) = X κ<γ tA κ ( P ′ ,s ′ ) ( t ) + X κ>γ A κ ( P ′ ,s ′ ) ( t ) , Where P ′ = A ⊕ · · · ⊕ A m , and where s ′ is the restrictions to P ′ . Hence the case p = 1 follows by induction (over m ) since this recursion preserves the interlacingproperty, see [21, Theorem 2.3].The case m > p > p just as for the case m = 1, n > { A γ ( P,s ) ( t ) } γ is an interlacing sequence, and thus A ( P,s ) ( t ) = X γ A γ ( P,s ) ( t ) , is real-rooted by e.g., [21, Theorem 2.3]. (cid:3) Next we address (b). A palindromic polynomial g ( t ) = a + a t + · · · + a n t n maybe written uniquely as g ( t ) = ⌊ d/ ⌋ X k =0 γ k ( g ) t k (1 + t ) d − k , where { γ k ( g ) } ⌊ d/ ⌋ k =0 are real numbers. If γ k ( g ) ≥ k , then we say that g ( t )is γ - positive , see [11]. Note that if g ( t ) is γ -positive, then { a i } ni =0 is a unimodalsequence , i.e., there is an index m such that a ≤ · · · ≤ a m ≥ a m +1 ≥ · · · ≥ a n . Conjecture 5.3. Suppose P is a sign-ranked labeled poset with nonnegative rankfunction ρ and s = ρ + 1 , then A ( P,s ) ( t ) is γ -positive.Remark . Let P be a sign-ranked labeled poset with a rank function ρ withvalues only in { , } , and let s = ρ + 1. Following the proof of [10, Theorem 4.2],with the use of Theorem 5.2, it follows that Conjecture 5.3 holds for ( P, s ). Weomit the technical details in recalling the proof here.If P is a naturally labeled ranked poset and s = ρ + 1, then O ( P, s ) is a closedintegral polytope and A ( P,s ) ( t ) is the so called h ∗ - polynomial of O ( P, s ). If the fol-lowing conjecture is true, then the coefficients of A ( P,s ) ( t ) form a unimodal sequenceby a powerful theorem of Bruns and R¨omer [8, Theorem 1]. Conjecture 5.4. Suppose P is a naturally labeled ranked poset, and let s = ρ + 1 .Then O ( P, s ) (or some related polytope with the same Ehrhart polynomial) has aregular and unimodular triangulation. Applications In this section we derive some applications of the generating function identitiesobtained in Section 3. If α = ( α , . . . , α p ) is a sequence, let | α | = α + · · · + α p .For τ = ( π, r ) ∈ L ( P, s ), letcomaj( τ ) = X i ∈ D ( τ ) p − i, andlhp( τ ) = | r | + X i ∈ D ( τ ) s ( π i +1 ) + · · · + s ( π p ) Theorem 6.1. If P is a labeled poset and s : [ p ] → Z + , then X n ≥ X f ∈ N ≤ n ( P,s ) q | r ( f ) | u | q ( f ) | t n = X τ ∈L ( P,s ) q | r | u comaj( τ ) t des s ( τ ) p Y i =0 (1 − u i t ) . (11) Proof. Set x i = u and y i = q for all 1 ≤ i ≤ p in (5). Then X τ ∈L ( P,s ) y r Y i ∈ D ( τ ) x π i +1 · · · x π p Y i ∈ [ p ] (1 − x π i · · · x π p t ) t | D ( τ ) | − t = X τ ∈L ( P,s ) q | r | u comaj( τ ) t des s ( τ ) Y i ∈ [ p ] (1 − tu p +1 − i )(1 − t )= X τ ∈L ( P,s ) q | r | u comaj( τ ) t des s ( τ ) Y i ∈ [ p ] (1 − tu i )(1 − t ) . The theorem follows. (cid:3) Theorem 6.2. If P is a labeled poset and s : [ p ] → Z + , then X n ≥ X f ∈ N ≤ n ( P,s ) q | f | t n = X τ ∈L ( P,s ) q lhp( τ ) t des s ( τ ) Y i ∈ [ p ] (cid:16) − tq P pj = i s ( π j ) (cid:17) (1 − t ) . (12) Proof. Set x i = q s ( i ) and y i = q for all 1 ≤ i ≤ p in (5). (cid:3) Corollary 6.3. If P is an anti-chain and s : [ p ] → Z + , then X n ≥ p Y i =1 ( u n + [ n ] u [ s ( i )] q ) t n = X τ ∈L ( P,s ) q | r | u comaj( τ ) t des s ( τ ) p Y i =0 (1 − u i t ) . (13) Proof. The corollary follows from Theorem 6.1 and Corollary 3.6. (cid:3) The wreath product of S p with a cyclic group of order k has elements Z k ≀ S p = { ( π, r ) : π ∈ S p and r : [ p ] → Z k } . The elements of Z k ≀ S p are often thought of as r -colored permutations. We mayidentify Z k ≀ S p with L ( P, s ) where P is an anti-chain on [ p ] and s ( i ) = k for all k ∈ [ p ]. For τ = ( π, r ) ∈ Z k ≀ S p definefmaj( τ ) = | r | + k · comaj( τ ) . Note that lhp( τ ) agrees with fmaj( τ ) when s = ( k, k, . . . , k ) . Below we derive a Carlitz formula for Z k ≀ S p first proved by Chow and Mansourin [12]. Corollary 6.4. For positive integers p and k , X n ≥ [ kn + 1] pq t n = X τ ∈ Z k ≀ S p t des s ( τ ) q fmaj( τ ) p Y i =0 (cid:0) − tq ki (cid:1) . (14) Proof. Let s = ( k, k, . . . , k ) and set u = q k in (13). Then p Y i =1 ( u n + [ n ] u [ s ( i )] q ) = (cid:0) q nk + [ n ] q k [ k ] q (cid:1) p = (cid:18) q nk + q kn − q k − q k − q − (cid:19) p = [ nk + 1] pq . The right hand side follows since s ( i ) = k for all 1 ≤ i ≤ p , and thus we sum overall τ ∈ Z k ≀ S p . (cid:3) Remark . The definition of fmaj above differs from the definition of the flagmajor index fmaj r in [12]. By the change in variables q → q − and t → tq kp andby noting that [ kn + 1] pq t n is invariant under this change of variables we find thatthe two flag major indices have the same distribution. Corollary 6.5. For positive integers p and k , X n ≥ p Y i =1 (1 + n [ k ] q i ) t n = X τ ∈ Z k ≀ S p q r q r · · · q r p p t des s ( τ ) (1 − t ) p +1 . Proof. Let s = ( k, k, . . . , k ) and set x i = 1 for all 1 ≤ i ≤ p in the equation displayedin Corollary 3.6. (cid:3) Remark . Note that when q i ≥ ≤ i ≤ p , the polynomial n p Y i =1 (1 + n [ k ] q i )has all its zeros in the interval [ − , X τ ∈ Z k ≀ S p q r q r · · · q r p p t des s ( τ ) is real-rooted in t . This generalizes [7, Theorem 6.4], where the case k = 2 wasobtained. References [1] M. Beck, B. Braun, Euler-Mahonian statistics via polyhedral geometry , Adv. in Math. (2013), 925–954[2] M. Beck, B. Braun, M. Koeppe, C. Savage, Z. Zafeirakopoulos, s -Lecture hall partitions,self-reciprocal polynomials, and Gorenstein cones , Ramanujan J., (1), (2015), 123–147.[3] M. Bousquet-Melou, K. Eriksson, Lecture hall partitions , Ramanujan J., (1), (1997), 101–111. [4] M. Bousquet-Melou, K. Eriksson, Lecture hall partitions, 2 , Ramanujan J., (2) (1997),165–185.[5] M. Bousquet-Melou, K. Eriksson, A Refinement of the Lecture Hall Theorem , Journal ofCombinatorial Theory, Series A (1999), 63–84.[6] F. Brenti, Unimodal, log-concave and P´olya frequency sequences in combinatorics , Mem.Amer. Math. Soc. (1989).[7] P. Br¨and´en, On linear transformations preserving the Polya frequency property , Transactionsof the American Mathematical Society , no. 8, (2006), 3697–3716.[8] W. Bruns, T. R¨omer, h -vectors of Gorenstein polytopes , J. Combin. Theory Ser. A, (1),(2007), 65–76.[9] P. Br¨and´en, Counterexamples to the Neggers-Stanley conjecture , Electron. Res. Announc.Amer. Math. Soc. (2004) 155–158.[10] P. Br¨and´en, Sign-graded posets, unimodality of W -polynomials and the Charney-Davis con-jecture, Electron. J. Combin. (2), (2004), Stanley Festschrift, R9.[11] P. Br¨and´en, Unimodality, log-concavity, real-rootedness and beyond , Handbook of Enumera-tive Combinatorics, 437–483, Discrete Math. Appl., CRC Press, Boca Raton, FL (2015).[12] C-O. Chow, T. Mansour, A Carlitz identity for the wreath product C r ≀ S n , Advances inApplied Mathematics , Issue 2 (2011), 199–215.[13] S. Corteel, S. Lee, C. D. Savage, Enumeration of sequences constrained by the ratio of con-secutive parts , S´em. Lothar. Combin., 54A:Art. B54Aa, 12 pp. (electronic), (2005/07).[14] M. Hyatt, Quasisymmetric functions and permutation statistics for Coxeter groups andwreath product groups, Ph.D. Thesis, University of Miami, (2011).[15] V. Reiner, V. Welker, On the Charney-Davis and Neggers–Stanley conjectures , J. Combin.Theory Ser. A (2), (2005) , 247–280.[16] T. Pensyl, C. D. Savage, Lecture hall partitions and the wreath products Z k ≀ S n , Integers,12B, Rational lecture hall polytopes and inflated Eulerian polynomials ,Ramanujan J., (2013), 97–114.[18] C. D. Savage, The mathematics of lecture hall partitions , Journal of Combinatorial Theory,Series A, (2016), 443–475.[19] C. D. Savage, M. J. Schuster, Ehrhart series of lecture hall polytopes and Eulerian polynomialsfor inversion sequences , Journal of Combinatorial Theory, Series A (2012), 850–870.[20] C. D. Savage, G. Viswanathan, The (1 /k ) -Eulerian Polynomials , Electr. J. Comb. (1): P9(2012).[21] C. D. Savage, M. Visontai, The s -Eulerian polynomials have only real roots, Trans. Amer.Math. Soc., (2), (2015), 1441–1466.[22] R. Stanley, Enumerative combinatorics, vol. I , Second edition, Cambridge University Press,2012.[23] R. Stanley, Enumerative combinatorics, vol. II , Cambridge University Press, 1999.[24] J. R. Stembridge, Counterexamples to the poset conjectures of Neggers, Stanley, and Stem-bridge , Trans. Amer. Math. Soc. (2007) 1115–1128.[25] D. G. Wagner, Enumeration of functions from posets to chains , European J. Combin. (1992), 313–324.[26] D. G. Wagner, Total positivity of Hadamard products, J. Math. Anal. Appl. (1992), no.2, 459–483. Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm,Sweden E-mail address : [email protected] Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden E-mail address ::