The binary diagonal X-ray and tournament score-vector conjecture

A binary diagonal X-ray of an $n \times n$ permutation matrix is its diagonal X-ray with entries in $\{0,1\}$. A tournament is a loopless directed graph in which exactly one of $(i,j)$ and $(j,i)$ is an arc for every pair of distinct vertices; its score vector is the nondecreasing vector of vertex out-degrees. The binary diagonal X-ray and tournament score-vector conjecture. The number of binary diagonal X-rays of $n \times n$ permutation matrices equals the number of score vectors of tournaments of order $n$. The conjecture connects two apparently equinumerous classes whose structural relationship is not clear; an equivalent characterization in terms of Toeplitz characteristics is given in the surrounding discussion.