Abstract
In two recent papers, Feigin proved that the Poincaré polynomials of the degenerate flag varieties have a combinatorial interpretation through Dellac configurations, and related them to the $q$-extended normalized median Genocchi numbers $\bar{c}_n(q)$ introduced by Han and Zeng, mainly by geometric considerations. In this paper, we give combinatorial proofs of these results by constructing statistic-preserving bijections between Dellac configurations and two other combinatorial models of $\bar{c}_n(q)$.
Highlights
The Genocchi numbers (G2n)n 1 = (1, 1, 3, 17, . . .) [13] and the median Genocchi numbers (H2n+1)n 0 = (1, 2, 8, 56, . . .) [14] can be defined as the positive integers G2n = g2n−1,n and H2n+1 = g2n+2,1 in the Seidel triangle1 j i defined by g2p−1,j = g2p−1,j−1 + g2p,j, g2p,j = g2p−1,j + g2p,j+1, with g1,1 = 1 and gi,j = 0 whenever i < j.the electronic journal of combinatorics 21(2) (2014), #P2.32 i\j 1 ↓It is well known that H2n+1 is divisible by 2n for all n 0
In two recent papers, Feigin proved that the Poincare polynomials of the degenerate flag varieties have a combinatorial interpretation through Dellac configurations, and related them to the q-extended normalized median Genocchi numbers cn(q) introduced by Han and Zeng, mainly by geometric considerations
We denote by Sn the set of permutations of the set [n] := {1, 2, . . . , n}, and by inv(σ) the number of inversions of a permutation σ ∈ Sn, i.e., the number of pairs (i, j) ∈ [n]2 with i < j and σ(i) > σ(j)
Summary
By introducing the subset Gn ⊂ Dn of normalized Genocchi permutations and using the combinatorial interpretation provided by Theorem 1.1, Han and Zeng proved combinatorially that the polynomial (1+q)n−1 divides Cn(1, q), which gives birth to polynomials (cn(q))n 1 defined by cn(q) = Cn(1, q)/(1 + q)n−1 This divisibility had previously been proved in the same paper with a continued fraction approach, as a corollary of the following theorem and a well-known result on continued fractions (see [8]). In [6, 7], Feigin introduced a q-analog of the normalized median Genocchi number hn with the Poincare polynomial PFna(q) of the degenate flag variety Fna (whose Euler characteristic is PFna(1) = hn), and gave a combinatorial interpretation of PFna(q) through Dellac configurations.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
