Bijective proofs for Eulerian numbers of types B and D

TL;DR

Bijective proofs of the identity Bn(t ) = (1 + t)Sn(t)− 2tSn( t) and of Stembridge’s identity are given.

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 $$B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2^n tS_n(t^2)$$ 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.

Figures (6)

• Figure 1: Signed permutations as paths and as barred permutations
• Figure 2: Negative inversions of $\overline{2}316\overline{4}\overline{7}5$, indexed on the left by the abscissa and ordinate and on the right by $\lambda^{u}_{\mathtt{y}},\lambda^{u}_{\mathtt{x}}$
• Figure 3: Characterizing inversions of the form $(i,j)$ with $i$ negative
• Figure 4: Two pairs of mates, the smooth mates are on the left
• Figure 5: The (unlabelled) graphs $2K_{2}$, $P_{3}$, and $C_{4}$

Theorems & Definitions (72)

• lemma 1
• proof
• definition 1
• proposition 1
• proposition 2
• proof
• remark 1
• definition 2
• definition 3
• lemma 2