Equidistributed Statistics on Fishburn Matrices and Permutations

Recently, Jelinek conjectured that there exists a bijection between certain restricted permutations and Fishburn matrices such that the bijection verifies the equidistribution of several statistics. The main objective of this paper is to establish such a bijection.

A note on the system of linear recurrence equations

We will find a solution to a system of 2d linear recurrence equations. Each equation is of the form x2k(n+1)=xk(n) or x2k+1(n + 1)=xk(n)+x2d-1+k(n). This kind of system is connected with counting restricted permutations.

Cycles and sorting index for matchings and restricted permutations

International audience We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters.