Counting Baxter Matrices

TL;DR

This paper studies Baxter matrices, showing their counts follow polynomial patterns for fixed rows and proving a conjecture about the maximum number of 1's based on matrix dimensions.

Contribution

It establishes polynomial formulas for counting Baxter matrices with fixed rows and proves a conjecture on the maximum number of 1's in such matrices.

Findings

Number of Baxter matrices with fixed rows is polynomial in columns.

Number of 1's in a Baxter matrix is less than the sum of rows and columns.

Polynomial degree depends on the number of rows.

Abstract

Donald Knuth recently introduced the notion of a Baxter matrix, generalizing Baxter permutations. We show that for fixed number of rows, $r$, the number of Baxter matrices with $r$ rows and $k$ columns eventually satisfies a polynomial in $k$ of degree $2r-2$. We also give a proof of Knuth's conjecture that the number of 1's in a $r \times k$ Baxter matrix is less than $r+k$.