Relatives of the Birkhoff polytope

A doubly-stochastic matrix is an n×n-matrix with nonnegative real entries, such that every row and column sums to 1. The set Bn of all such n×n-matrices is a nice convex object, called the n'th Birkhoff polytope. It's a hard and wide-open problem to compute the volume of Bn.There are various relatives of these polytopes; here is one example:Instead of n-by-n permutation matrices, consider alternating-sign matrices. Their convex hull is a polytope which was recently studied by J. Striker.We will explore if anything be said about the volumes of these polytopes, e.g., some analogues of Canfield-McKay's asymptotic formula for the volume of Bn.

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