# 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 = ⌈ (nā2)/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} ā 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} ā 𝒜{n},
𝔅{±}{n} and 𝔇{±}{n}. As a corollary, we get moment type
identities for coefficients of such polynomials.

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