Combinatorial Proofs of Identities Involving Symmetric Matrices

TL;DR

This paper provides combinatorial proofs for identities involving symmetric matrices and involutions, using the Robinson-Schensted-Knuth correspondence, and offers a streamlined proof of Brualdi and Ma's main theorem.

Contribution

It introduces combinatorial proofs for algebraic identities related to symmetric matrices and involutions, utilizing the RSK correspondence to simplify existing proofs.

Findings

Combinatorial proofs of identities involving symmetric matrices

Restatement and simplification of Brualdi and Ma's main theorem

Application of RSK correspondence to involutions and matrices

Abstract

Brualdi and Ma found a connection between involutions of length $n$ with $k$ descents and symmetric $k\times k$ matrices with non-negative integer entries summing to $n$ and having no row or column of zeros. From their main theorem they derived two alternating sums by algebraic means and asked for combinatorial proofs. In this note we provide such demonstrations making use of the Robinson-Schensted-Knuth correspondence between symmetric matrices and semi-standard Young Tableau. Additionally, we restate the proof of Brualdi and Ma's main result with this perspective which shortens the argument.