Eli Bagno

Block decomposition of permutations and Schur-positivity

The block number of a permutation is the maximum number of components in its expression as a direct sum. We show that, for 321-avoiding permutations, the set of left-to-right maxima has the same distribution when the block number is assumed to be k, as when the last descent of the inverse is assumed to be at position

ON THE EXCEDANCE NUMBER OF COLORED PERMUTATION GROUPS

n - k

Euler-Mahonian parameters on colored permutation groups.

n-k. This result is analogous to the Foata–Schützenberger equidistribution theorem, and implies that the quasi-symmetric generating function of the descent set over 321-avoiding permutations with a prescribed number of blocks is Schur-positive.

Statistics on the multi-colored permutation groups

Abstract The excedance number for Sn is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct proof based on a recursion which uses only excedances and extend it to the flag-excedance parameter defined on the group of colored permutations Gr,n = ℤr ≀ Sn. We have also computed the distribution of a variant of the flag-excedance number, and show that its enumeration uses the Stirling number of the second kind. Moreover, we show that the generating function of the flag-excedance number defined on ℤr ≀ Sn is symmetric, and its variant is log-concave on ℤr ≀ Sn..