Bijective proofs for Eulerian numbers of types B and D

Abstract

Let $\Bigl\langle\matrix{n\cr k}\Bigr\rangle$, $\Bigl\langle\matrix{B_n\cr k}\Bigr\rangle$, and $\Bigl\langle\matrix{D_n\cr k}\Bigr\rangle$ 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 $S_n(t) = \sum_{k = 0}^{n-1} \Bigl\langle\matrix{n\cr k}\Bigr\rangle t^k$, $B_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{B_n\cr k}\Bigr\rangle t^k$, and $D_n(t) = \sum_{k = 0}^n \Bigl\langle\matrix{D_n\cr k}\Bigr\rangle t^k$. We give bijective proofs of the identity $$B_n(t^2)=(1+t{)}^{n+1}{S}_n(t)-2^{n}t{S}_n(t^2)$$ and of Stembridge's identity $$D_n(t)=B_n(t)-n{2}^{n-1}t{S}_{n-1}(t).$$ 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.

Discrete Mathematics and Theoretical Computer Science, 2023