Equidistribution and Sign-Balance on 321-Avoiding Permutations

Let $T_n$ be the set of 321-avoiding permutations of order $n$. Two properties of $T_n$ are proved: (1) The {\em last descent} and {\em last index minus one} statistics are equidistributed over $T_n$, and also over subsets of permutations whose inverse has an (almost) prescribed descent set. An analogous result holds for Dyck paths. (2) The sign-and-last-descent enumerators for $T_{2n}$ and $T_{2n+1}$ are essentially equal to the last-descent enumerator for $T_n$. The proofs use a recursion formula for an appropriate multivariate generating function.Comment: 17 pages; to appear in S\'em. Lothar. Combi

On the distribution of some Euler-Mahonian statistics

We give a direct combinatorial proof of the equidistribution of two pairs of permutation statistics, (des, aid) and (lec, inv), which have been previously shown to have the same joint distribution as (exc, maj), the major index and the number of excedances of a permutation. Moreover, the triple (pix, lec, inv) was shown to have the same distribution as (fix, exc, maj), where fix is the number of fixed points of a permutation. We define a new statistic aix so that our bijection maps (pix, lec, inv) to (aix, des, aid). We also find an Eulerian partner das for a Mahonian statistic mix defined using mesh patterns, so that (das, mix) is equidistributed with (des, inv).Comment: 9 page