Sharper bounds for zero-one matrices with prescribed margins

Refine the upper bound on the number of zero-one matrices with specified row and column margins by using the multivariate Kummer carry structure underlying Algorithm 1.

Background

A corollary of Section 5 of Bernard–Letac gives an upper bound for the number of zero-one matrices with prescribed margins. The paper observes that the carry information computed by the p-adic valuation algorithm may provide additional structure for improving this estimate.

The problem is connected with the longstanding counting challenge identified by Ryser and is explicitly left unresolved.