The Preisach graph and longest increasing subsequences

TL;DR

This paper establishes a bijection between vertices of the Preisach graph and increasing subsequences of a permutation, linking the nesting degree of the graph to the length of the longest increasing subsequence.

Contribution

It introduces an explicit bijection connecting Preisach graph vertices with increasing subsequences, revealing the nesting degree equals the longest increasing subsequence length.

Findings

Nesting degree of the Preisach graph equals the length of the longest increasing subsequence.

Established a bijection between graph vertices and increasing subsequences.

Connected graph hierarchy structure with permutation subsequence properties.

Abstract

The Preisach graph is a directed graph associated with a permutation $\rho\in{\cal S}_N$. We give an explicit bijection between its vertices and increasing subsequences of $\rho$ with the property that the length of a subsequence equals to the degree of nesting of the corresponding vertex inside a hierarchy of cycles and sub-cycles of the graph. As a consequence, the nesting degree of the Preisach graph equals the length of the longest increasing subsequence.