A Mysterious Cluster Expansion Associated to the Expectation Value of the Permanent of 0–1 Matrices

Abstract

We consider two ensembles of 0 -1 n × n matrices. The first is the set of all n × n matrices with entries zeroes and ones such that all column sums and all row sums equal r, uniformly weighted. The second is the set of n × n matrices with zero and one entries where the probability that any given entry is one is r/n, the probabilities of the set of individual entries being i.i.d.'s. Calling the two expectation values E and EB respectively, we develop

On the Approximate Asymptotic Statistical Independence of the Permanents of 0-1 Matrices

“…Pernici has shown me how to alternatively derive these equations using the formalism of [4], assuming Conjecture 1 therein. )…”

“…These same calculations show (1.3) is not true with O 1/n 4 replaced by O 1/n 5 . E 1 is often used as an 'approximation' to E. In [1] and [4] certain aesthetic relations seem to hold in the same form for E and E 1 expectations. Similarly we believe eq (1.1) presages the truth of eq (1.3).…”

Rank relations between a {0, 1}-matrix and its complement

“…{ , }-matrices are closely related to graph theory and combinatorial mathematics [1][2][3][4]. They also have a wide range of practical applications in statistics and probability [5][6][7][8].…”