10 problems

Diaconis–Eriksson conjecture. The ideals of the Birkhoff polytopes are generated in degree $3$.

Let $P$ be the Birkhoff polytope, whose points are the real doubly stochastic $n\times n$ matrices, and let $E(P)$ be the affine monoid encoding the magic squares. Question (ii) as…

The real-rootedness and log-concavity conjecture. One has

The determinant factorization conjecture. Writing $n=2r$ or $n=2r-1$, respectively,

For a positive integer $n$, let $d_n(0,t)$ satisfy a recursion … where each $c_i(t)$ is a polynomial in $t$, and let $t_0$ be the maximum integer root, if one exists, of…

For a positive integer $n$, let $H_n(t)$ be the number of nonnegative integral $n\times n$ matrices whose row sums and column sums all equal $t$. The generating function has the fo…

Let the Birkhoff polytope be the polytope of doubly stochastic matrices, and call an integral polytope Ehrhart positive when every coefficient of its Ehrhart polynomial is nonnegat…

The Birkhoff polytope $B_n$ is the convex hull of all $n\times n$ permutation matrices, equivalently the set of real nonnegative $n\times n$ matrices whose row and column sums are…

A face of the Birkhoff polytope is a face of the convex polytope of doubly stochastic matrices. Two faces are combinatorially isomorphic when their face lattices are isomorphic, an…

Birkhoff-polytope uniqueness conjecture. If $P(G)$ is combinatorially equivalent to $B_n$ for some $n$, then $G$ is effectively equivalent to $S_n$.