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We consider two ensembles of nxn matrices. The first is the set of all nxnmatrices with entries zeroes and ones such that all column sums and all rowsums equal r, uniformly weighted. The second is the set of nxn matrices withzero 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 thetwo expectation values E and EB respectively, we develop a formal relationE(perm(A)) = EB(perm (A)) x exp{sum Ti}. We also use a well-known approximatingensemble to E, E1. We prove using E or E1 one obtains the same value of Ti fori < 21. (THE PUBLISHED VERSION OF THIS PAPER ONLY OBTAINS RESULTS FOR i < 8. Wego beyond the results of the published version by taking much more advantage ofrecent work of Pernici and of Wanless on i-matchings on regular bipartitegraphs.)These terms Ti, i < 21, have amazing properties. We conjecture thatthese properties hold also for all i.

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