On the alternating runs polynomial in type B and type D Coxeter groups

Dey, HK and Sivasubramanian, S (2023) On the alternating runs polynomial in type B and type D Coxeter groups. In: Journal of the Ramanujan Mathematical Society, 38 (3). pp. 215-223.

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Abstract

Let Rn(t) denote the polynomial enumerating alternating runs in the symmetric group Sn. Wilf showed that (1+t)m divides Rn(t) where m = �(n�2)/2�. Bóna recently gave a group-action-based proof of this fact. In this work, we give a group-action-based proof for type B and type D analogues of this result. Interestingly, our proof gives a group action on the positive/negative parts B±n and D±n and so we get refinements of the result to the case when summation is over B±n and D±n . We are unable to get a group-action-based proof of Wilf�s result when summation is over the alternating group An and over Sn�An, but using other ideas, give a different proof. We give similar results to the polynomial which enumerates alternating sequences in An, Sn�An, B±n and D±n . As a corollary, we get moment type identities for coefficients of such polynomials. © 2023 Ramanujan Mathematical Society. All rights reserved.

Item Type:	Journal Article
Publication:	Journal of the Ramanujan Mathematical Society
Publisher:	Ramanujan Mathematical Society
Additional Information:	The copyright for this article belongs toRamanujan Mathematical Society.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	29 Feb 2024 11:41
Last Modified:	29 Feb 2024 11:41
URI:	https://eprints.iisc.ac.in/id/eprint/83812

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