On minimal pattern-containing inversion sequences

TL;DR

This paper introduces minimal inversion sequences for patterns, characterizes them, and explores their enumeration, including connections to poly-Bernoulli numbers, advancing understanding of pattern avoidance in inversion sequences.

Contribution

It defines and characterizes minimal inversion sequences, providing bounds, enumeration results, and links to combinatorial structures like increasing trees and poly-Bernoulli numbers.

Findings

Characterization of $ ho$-minimal inversion sequences

Enumeration of minimal inversion sequences for specific patterns

Connection between inversion sequences equal to their reduction and poly-Bernoulli numbers

Abstract

We introduce the notion of minimal inversion sequences for a pattern $\rho$, which form the smallest set of inversion sequences whose avoidance is equivalent to the avoidance of $\rho$ for inversion sequences. We give a characterization of $\rho$-minimal inversion sequences based on the occurrences of the pattern $\rho$ they contain, and use it to find upper and lower bounds on the lengths of $\rho$-minimal inversion sequences. We provide some enumerative results on the exact number of minimal inversion sequences for some patterns, through a bijection with increasing trees, and some exhaustive generation. Lastly, we enumerate inversion sequences which are equal to their reduction, and find an interesting connection with poly-Bernoulli numbers.