A note on Diagonal sequences of integer partitions

TL;DR

This paper investigates the structure of partitions of integers based on their diagonal sequences, revealing a partial order with extremal elements and providing formulas for counting partitions sharing the same diagonal sequence.

Contribution

It introduces a novel analysis of diagonal sequences of partitions, establishing a partial order and explicit enumeration formulas for partitions with identical diagonal sequences.

Findings

Partitions with the same diagonal sequence form a partially ordered set.

There are unique maximal and minimal partitions within each diagonal sequence class.

Explicit formulas for counting partitions with a given diagonal sequence are provided.

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.