Séminaire Lotharingien de Combinatoire, B90a (2026), 16 pp.

A Bijective Proof of an Identity of Berkovich and Uncu

Aritram Dhar and Avi Mukhopadhyay

Abstract. The BG-rank BG(π) of an integer partition π is defined as BG(π) := i-j, where i is the number of odd-indexed odd parts and j is the number of even-indexed odd parts of π. In a recent work, Fu and Tang ask for a direct combinatorial proof of the following identity of Berkovich and Uncu:
[ ] B (k,q) = q2k2-k 2N + ν , 2N+ ν N + k q2
where, for any integer k, any nonnegative integer N, and ν ∈ {0,1}, BN(k,q) is the generating function for partitions into distinct parts less than or equal to N with BG-rank equal to k, and [ ] a+bbq is the Gaussian binomial coefficient. In this paper, we provide a bijective proof of Berkovich and Uncu's identity along the lines of Vandervelde and Fu and Tang's idea.

Received: December 16, 2023. Revised: September 10, 2024. Accepted: September 10, 2024.

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