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arXiv · 2207.10252

Statistics of Partial Permutations via Catalan matrices

Yen-Jen Cheng·Sen-Peng Eu·Hsiang-Chun Hsu
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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.

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Yen-Jen Cheng, Sen-Peng Eu, Hsiang-Chun Hsu. 2022-07-21. Statistics of Partial Permutations via Catalan matrices. https://arxiv.org/abs/2207.10252

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