## Abstract

A generalized Catalan matrix (a_{n,k})_{n,k≥0} is generated by two seed sequences s=(s_{0},s_{1},…) and t=(t_{1},t_{2},…) together with a recurrence relation. By taking s_{ℓ}=2ℓ+1 and t_{ℓ}=ℓ^{2} we can interpret a_{n,k} as the number of partial permutations, which are n×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.

Original language | English |
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Article number | 102451 |

Journal | Advances in Applied Mathematics |

Volume | 143 |

DOIs | |

Publication status | Published - 2023 Feb |

## Keywords

- Catalan matrix
- Connected permutation
- Cycle
- Cycle-up-down permutation
- Descent
- Excedance
- Fixed point
- Inversion
- Partial permutation
- Permutation
- Right-to-left minimum
- Set-valued statistic
- Statistic

## ASJC Scopus subject areas

- Applied Mathematics