Bregman's Theorem

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Friday, 22 February 2013, 14:30 to 16:00

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Abstract: Let A be an $n \times n$ boolean matrix, i.e., its entries come from the set $\{0,1\}$. Let $r_i$ denote the number of 1's in the row $i$. Also, let $S$ denote the set of permutations, $\sigma$, on $[n]$ such that $A_{i, \sigma(i)}=1$for all $i \in [n]$. Then the $perm(A)=|S|$. Bregman's theorem is stated as follows:

$$perm(A) \leq \prod_{i\in [n]}(r_i !)^{1/r_i}$$

This upper bounds the number of perfect matchings in a bi-partite graph. We are going to prove it.