Journal of the Ramanujan Mathematical Society

Volume 38, Issue 3, September 2023 pp. 215–223.

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

Authors: Hiranya Kishore Dey and Sivaramakrishnan Sivasubramanian
Author institution: Department of Mathematics, Indian Institute of Science, Bangalore, Bangalore 560 012, India.

Summary: Let R{n}(t) denote the polynomial enumerating alternating runs in the symmetric group 𝔖{n}. Wilf showed that (1+t){m} divides R{n}(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 𝒜{n} and over 𝔖{n} − 𝒜{n}, but using other ideas, give a different proof. We give similar results to the polynomial which enumerates alternating sequences in 𝒜{n}, 𝔖{n} − 𝒜{n}, 𝔅{±}{n} and 𝔇{±}{n}. As a corollary, we get moment type identities for coefficients of such polynomials.

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