On restricted permutations on regular multisets

TL;DR

This paper extends pattern avoidance analysis to permutations on regular multisets, providing complete enumeration formulas and exploring connections to well-known sequences and conjectures.

Contribution

It completes the enumeration of permutations avoiding pairs of length-three patterns on regular multisets, filling a gap in prior research.

Findings

Closed enumeration formulas for all studied cases

Identification of well-known sequences in enumeration results

Discussion on generalizing the Stanley-Wilf conjecture

Abstract

The extension of pattern avoidance from ordinary permutations to those on multisets gave birth to several interesting enumerative results. We study permutations on regular multisets, i.e., multisets in which each element occurs the same number of times. For this case, we close a gap in the work of Heubach and Mansour (2006) and complete the study of permutations avoiding a pair of patterns of length three. In all studied cases, closed enumeration formulae are given and well-known sequences appear. We conclude this paper by some remarks on a generalization of the Stanley-Wilf conjecture to permutations on multisets and words.