A note on Diagonal sequences of integer partitions

Michael Neubauer; Harmony Vargas

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arXiv · 2412.06856

A note on Diagonal sequences of integer partitions

Michael Neubauer·Harmony Vargas

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Abstract

Let $\mathcal{P}(n)$ be the set of partitions of the positive integer $n$. For $\alpha=(\alpha_1,...,\alpha_t) \in \mathcal{P}(n)$ define the diagonal sequence $\delta(\alpha)=(d_k(\alpha))_{k \geq 1}$ via $ d_k(\alpha) = \big\lvert \{ i \, \rvert \, 1 \leq i \leq k \mbox{ and } \alpha_i + i- 1\geq k \} \big\rvert.$ We show that the set of all partitions in $\mathcal{P}(n)$ with the same diagonal sequence is a partially ordered set under majorization with unique maximal and minimal elements and we give an explicit formula for the number of partitions with the same diagonal sequence.

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Michael Neubauer, Harmony Vargas. 2024-12-08. A note on Diagonal sequences of integer partitions. https://arxiv.org/abs/2412.06856

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