The excedances and descents of bi-increasing permutations

Starting from some considerations we make about the relations between certain difference statistics and the classical permutation statistics we study permutations whose inversion number and excedance difference coincide. It turns out that these (so-called bi-increasing) permutations are just the 321-avoiding ones. The paper investigates their excedance and descent structure. In particular, we find some nice combinatorial interpretations for the distribution coefficients of the number of excedances and descents, respectively, and their difference analogues over the bi-increasing permutations in terms of parallelogram polyominoes and 2-Motzkin paths. This yields a connection between restricted permutations, parallelogram polyominoes, and lattice paths that reveals the relations between several well-known bijections given for these objects (e.g. by Delest-Viennot, Billey-Jockusch-Stanley, Francon-Viennot, and Foata-Zeilberger). As an application, we enumerate skew diagrams according to their rank and give a simple combinatorial proof for a result concerning the symmetry of the joint distribution of the number of excedances and inversions, respectively, over the symmetric group.Comment: 36 page

On the diagram of 132-avoiding permutations

The diagram of a 132-avoiding permutation can easily be characterized: it is simply the diagram of a partition. Based on this fact, we present a new bijection between 132-avoiding and 321-avoiding permutations. We will show that this bijection translates the correspondences between these permutations and Dyck paths given by Krattenthaler and by Billey-Jockusch-Stanley, respectively, to each other. Moreover, the diagram approach yields simple proofs for some enumerative results concerning forbidden patterns in 132-avoiding permutations.Comment: 20 pages; additional reference is adde

The area above the Dyck path of a permutation

In this paper we study a mapping from permutations to Dyck paths. A Dyck path gives rise to a (Young) diagram and we give relationships between statistics on permutations and statistics on their corresponding diagrams. The distribution of the size of this diagram is discussed and a generalisation given of a parity result due to Simion and Schmidt. We propose a filling of the diagram which determines the permutation uniquely. Diagram containment on a restricted class of permutations is shown to be related to the strong Bruhat poset.Comment: 9 page

Refined sign-balance on 321-avoiding permutations

AbstractThe number of even 321-avoiding permutations of length n is equal to the number of odd ones if n is even, and exceeds it by the n−12th Catalan number otherwise. We present an involution that proves a refinement of this sign-balance property respecting the length of the longest increasing subsequence of the permutation. In addition, this yields a combinatorial proof of a recent analogous result of Adin and Roichman dealing with the last descent. In particular, we answer the question of how to obtain the sign of a 321-avoiding permutation from the pair of tableaux resulting from the Robinson–Schensted–Knuth algorithm. The proof of the simple solution is based on a matching method given by Elizalde and Pak