Bijective proofs for Eulerian numbers of types B and D
Résumé
Let $< n,k> $, $< B_n k> $, and $< D_n, k >$ be the Eulerian numbers in the types A, B, and D, respectively---that is, the number of permutations of n elements with k descents, the number of signed permutations (of n elements) with k type B descents, the number of even signed permutations (of n elements) with k type D descents. Let $S_n(t) = \sum_{k = 0}^{n-1} < n,k> t^k$, $B_n(t) = \sum_{k = 0}^{n}< B_n,k >t^k$, and $D_n(t) = \sum_{k = 0}^{n}< D_n,k> t^k$. We give bijective proofs of the identity $B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2^{n}tS_n(t^2)$ and of Stembridge's identity $D_n(t) = B_n (t) - n2^{n−1}tS_{n−1}(t)$. These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs $(w, E)$ with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$, which we use to obtain bijective proofs of enumerative results for threshold graphs.
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