Gamma Positivity of the Excedance-Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups
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Annals of Combinatorics
Gamma Positivity of the Excedance-Based Eulerian Polynomial in Positive Elements of Classical Weyl Groups Hiranya Kishore Dey and Sivaramakrishnan Sivasubramanian Abstract. The Eulerian polynomial AExcn (t) enumerating excedances in the symmetric group Sn is known to be gamma positive for all n. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also gamma positive for all n. We − consider AExc+ n (t) and AExcn (t), the polynomials which enumerate excedance in the alternating group An and in Sn − An , respectively. We show that AExc+ n (t) is gamma positive iff n ≥ 5 is odd. When n ≥ 4 is even, AExc+ n (t) is not even palindromic, but we show that it is the sum of two gamma positive summands. An identical statement is true about AExc− n (t). We extend similar results to the excedance based Eulerian polynomial when enumeration is done over the positive elements in both type B and type D Coxeter groups. Gamma positivity results are known when excedance is enumerated over derangements in Sn . We extend some of these to the case when enumeration is done over even and odd derangements in Sn . Mathematics Subject Classification. 05A05, 05A19, 05E15. Keywords. Gamma Classical Weyl Groups.
positivity,
Eulerian
polynomial,
1. Introduction For a positive integer n, let [n] = {1, 2, . . . , n} and let Sn be the set of permutations on [n]. For π = π1 , π2 , . . . , πn ∈ Sn , define its excedance set as EXC(π) = {i ∈ [n] : πi > i} and its number of excedances as exc(π) = |EXC(π)|. Define its number of non-excedances as nexc(π) = |{i ∈ [n] : πi ≤ i}|. For π ∈ Sn , define its number of inversions as inv(π) = |{1 ≤ i < j ≤ n : πi > πj }|. Let DES(π) = {i ∈ [n − 1] : πi > πi+1 } and ASC(π) = {i ∈ [n − 1] : πi < πi+1 } be its set of descents and ascents, respectively. Let des(π) = |DES(π)| be its 0123456789().: V,-vol
H. K. Dey et al.
number of descents and asc(π) = |ASC(π)| be its number of ascents. The polynomial An (t) = π∈Sn tdes(π) is the classical Eulerian polynomial. Let An ⊆ Sn be the subset of even permutations. Define An (t) =
tdes(π) and An (s, t) =
π∈Sn
AExcn (t) =
texc(π) and AExcn (s, t) =
texc(π) snexc(π)−1 ,
(2)
texc(π) snexc(π)−1 ,
(3)
π∈Sn
texc(π) and AExc+ n (s, t) =
π∈An
AExc− n (t) =
(1)
π∈Sn
π∈Sn
AExc+ n (t) =
tdes(π) sn−1−des(π) ,
π∈Sn −An
π∈An
texc(π) and AExc− n (s, t) =
texc(π) snexc(π)−1 .
π∈Sn −An
(4) It is a well-known result of MacMahon [11] that both descents and excedances are equidistributed over Sn . That is, for all positive integers n, An (t) = AExcn (t). n Let f (t) ∈ Q[t] be a degree n univariate polynomial with f (t) = i=0 ai ti where an = 0. Let r be the least non-negative integer such that ar = 0. Define len(f ) = n − r. The polynomial f (t) is said to be palindromic if ar+i = an−i for 0 ≤ i ≤ (n − r)/2. Define the center of symmetry of f (t) to be (n + r)/2. Note that for a palindromic polynomial f (t), its center of symmetry could be half integral. L
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