The Bernoulli numbers are closely related to alternating permutations. That is to say, permutations like $1524736$ 1524376$where the numbers alternately go up and down. Specifically, if$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. 1 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.