Eulerian numbers, tableaux, and the Betti numbers of a toric variety
Discrete Mathematics - Special volume: algebraic combinatorics
Counting permutations with given cycle structure and descent set
Journal of Combinatorial Theory Series A
On the Neggers-Stanley conjecture and the Eulerian polynomials
Journal of Combinatorial Theory Series A
Permutation statistics on involutions
European Journal of Combinatorics
Actions on permutations and unimodality of descent polynomials
European Journal of Combinatorics
The descent statistic on involutions is not log-concave
European Journal of Combinatorics
The symmetric and unimodal expansion of Eulerian polynomials via continued fractions
European Journal of Combinatorics
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Let In,k (respectively Jn,k) be the number of involutions (respectively fixed-point free involutions) of {1,...,n} with k descents. Motivated by Brenti's conjecture which states that the sequence In,0, In,1,..., In,n-1 is log-concave, we prove that the two sequences In,k and J2n,k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers an,k such that Σk=0n-1 In,ktk= Σk=0⌊(n-1)/2⌋ an,ktk(1+t)n-2k-1.This statement is stronger than the unimodality of In,k but is also interesting in its own right.