Statistics of partial permutations via Catalan matrices

Yen Jen Cheng*, Sen Peng Eu, Hsiang Chun Hsu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


A generalized Catalan matrix (an,k)n,k≥0 is generated by two seed sequences s=(s0,s1,…) and t=(t1,t2,…) together with a recurrence relation. By taking s=2ℓ+1 and t=ℓ2 we can interpret an,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 languageEnglish
Article number102451
JournalAdvances in Applied Mathematics
Publication statusPublished - 2023 Feb


  • 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


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