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The Worpitzky identity for the groups of signed and even-signed permutations

Abstract

The well-known Worpitzky identity

$$\begin{aligned} (x+1)^n = \sum \limits _{k=0}^{n-1} A_{n,k} {{x+n-k} \atopwithdelims (){n}} \end{aligned}$$

provides a connection between two bases of \(\mathbb {Q}[x]\): the standard basis \((x+1)^n\) and the binomial basis \({{x+n-k} \atopwithdelims (){n}}\), where the Eulerian numbers \(A_{n,k}\) for the symmetric group serve as the entries of the transformation matrix. Brenti has generalized this identity to the Coxeter groups of types \(B_n\) and \(D_n\) (signed and even-signed permutations groups, respectively) using generatingfunctionology. Motivated by Foata–Schützenberger’s and Rawlings’ proof for the Worpitzky identity in the symmetric group, we provide combinatorial proofs for the generalizations of this identity and for their q-analogues in the Coxeter groups of types \(B_n\) and \(D_n\). Our proofs utilize the language of P-partitions for the \(B_n\)- and \(D_n\)-posets, introduced by Chow and Stembridge, respectively.

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Acknowledgements

We want to thank Kyle Petersen for suggesting us to use the idea of P-partitions in this work and for granting us many fruitful ideas. We also would like to thank Yuval Roichman and Ron Adin for sharing with us the algebraic insight on Lemma 5.2, expressed in Remark 5.3. We also thank an anonymous referee for helpful remarks.

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Correspondence to David Garber.

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Eli Bagno and David Garber were partially supported by the Israeli Ministry of Science and Technology, and the French National Scientific Research Center (CNRS), Grant PRC 1656.

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Bagno, E., Garber, D. & Novick, M. The Worpitzky identity for the groups of signed and even-signed permutations. J Algebr Comb 55, 413–428 (2022). https://doi.org/10.1007/s10801-021-01056-4

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