Hertzsprung patterns on involutions

TL;DR

This paper studies Hertzsprung patterns, a special class of permutation subsequences, within involutions, providing formulas for counting their occurrences and analyzing their distributions for small lengths.

Contribution

It introduces a general enumeration formula for Hertzsprung patterns in involutions and analyzes their distributions for lengths two and three.

Findings

Derived a formula for counting Hertzsprung patterns in involutions.

Analyzed distributions of patterns of lengths two and three.

Provided insights into the structural constraints of involutions.

Abstract

Hertzsprung patterns, recently introduced by Anders Claesson, are subsequences of a permutation contiguous in both positions and values, and can be seen as a subclass of bivincular patterns. This paper investigates Hertzsprung patterns within involutions, where additional structural constraints introduce new challenges. We present a general formula for enumerating occurrences of these patterns in involutions. We also analyze specific cases to derive the distribution of all Hertzsprung patterns of lengths two and three.