Combinatorial identities related to $2\times 2$ submatrices of recursive matrices

TL;DR

This paper explores combinatorial identities involving sums of $2\times 2$ minors and permanents of recursive matrices, extending previous work and discovering new identities with Narayana polynomials using computer algebra tools.

Contribution

It generalizes existing identities related to recursive matrices and introduces new identities involving Narayana polynomials through computational methods.

Findings

Derived combinatorial identities for sums of minors and permanents.

Extended previous results by Sun and Ma on recursive matrices.

Discovered new identities involving Narayana polynomials.

Abstract

Recursive matrices are ubiquitous in combinatorics, which have been extensively studied. We focus on the study of the sums of $2\times 2$ minors of certain recursive matrices, the alternating sums of their $2\times 2$ minors, and the sums of their $2\times 2$ permanents. We obtain some combinatorial identities related to these sums, which generalized the work of Sun and Ma in [{\it Electron. J. Combin. 2014}] and [{\it European J. Combin. 2014}]. With the help of the computer algebra package {\tt HolonomicFunctions}, we further get some new identities involving Narayana polynomials.