Pattern-avoidance and Fuss-Catalan numbers

TL;DR

This paper explores permutations with entries sharing the same remainder as their position modulo k, revealing connections to Fuss-Catalan and Raney numbers, and extends these ideas to subexcedant functions and pattern-avoiding permutations.

Contribution

It introduces new classes of pattern-avoiding permutations with modular restrictions, linking them to Fuss-Catalan and Raney numbers, and provides complete enumeration results for specific pattern-avoidance cases.

Findings

Pattern-avoidance with mod k restrictions yields Fuss-Catalan numbers.

Mod k restrictions on subexcedant functions relate to classical combinatorial sequences.

Complete enumeration of mod-k-alternating permutations avoiding two patterns of length 3.

Abstract

We study a subset of permutations, where entries are restricted to having the same remainder as the index, modulo some integer $k \geq 2$. We show that when also imposing the classical 132- or 213-avoidance restriction on the permutations, we recover the Fuss--Catalan numbers and some special cases of the Raney numbers. Surprisingly, an analogous statement also holds when we impose the mod $k$ restriction on a Catalan family of subexcedant functions. Finally, we completely enumerate all combinations of mod-$k$-alternating permutations, avoiding two patterns of length 3. This is analogous to the systematic study by Simion and Schmidt, of permutations avoiding two patterns of length 3.