Inverse Relations in Shapiro's Open Questions

TL;DR

This paper explores the role of involutions with invariant sequences in combinatorics, specifically addressing open questions related to the Fibonacci matrix and Riordan involutions, providing new insights and solutions.

Contribution

It introduces the use of invariant sequences to answer open questions about the Fibonacci matrix and Riordan involutions in combinatorics.

Findings

Solved open questions about the Fibonacci matrix.

Provided new characterizations of Riordan involutions.

Enhanced understanding of involution properties in combinatorics.

Abstract

As an inverse relation, involution with an invariant sequence plays a key role in combinatorics and features prominently in some of Shapiro's open questions [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585-596]. In this paper, invariant sequences are used to provide answers to some of these questions about the Fibonacci matrix and Riordan involutions.