Counting Baxter Matrices

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 $r+k$.