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The Bernoulli numbers are closely related to alternating permutations. That is to say, permutations like $1524736$ where the numbers alternately go up and down. Specifically, if $A_n$ is the number of such permutations of an $n$ element set, then $$B_{2n} = (-1)^{n-1} \frac{2n}{4^{2n}-2^{2n}} A_{2n-1}.$$ It's possible you could somehow relate your sums over diagrams to alternating permutations.