Statistics of Partial Permutations via Catalan matrices

TL;DR

This paper explores the combinatorial properties of partial permutations through generalized Catalan matrices, connecting permutation statistics with seed sequences and analyzing specific permutation families.

Contribution

It introduces a framework linking partial permutation statistics to Catalan matrices and seed sequences, extending classical permutation analysis.

Findings

Partial permutations can be characterized using generalized Catalan matrices.

Most permutation statistics can be encoded via seed sequences in this framework.

Results include analysis of connected and cycle-up-down permutations.

Abstract

A generalized Catalan matrix $(a_{n,k})_{n,k\ge 0}$ is generated by two seed sequences $\mathbf{s}=(s_0,s_1,\ldots)$ and $\mathbf{t}=(t_1,t_2,\ldots)$ together with a recurrence relation. By taking $s_\ell=2\ell+1$ and $t_\ell=\ell^2$ we can interpret $a_{n,k}$ as the number of partial permutations, which are $n\times n$ $0,1$-matrices of $k$ zero rows with at most one $1$ in each row or column. In this paper we prove that most of fundamental statistics and some set-valued statistics on permutations can also be defined on partial permutations and be encoded in the seed sequences. Results on two interesting permutation families, namely the connected permutations and cycle-up-down permutations, are also given.