Proposal of a generating function of partition sequences

TL;DR

This work tackles constructing generating functions for partition sequences in a way that preserves partition-specific information, by using a fixed matrix substitution on the Maya-diagram representation $M(\lambda)$. The main idea converts a naive unit-coefficient sum over partitions into a matrix-weighted generating function with matrices $X$ and $Y$, enabling encoding of statistics such as $|\lambda|$, $\ell(\lambda)$, and $a(\lambda)$; for example, the general sum satisfies $\sum_{\lambda\in\mathcal{P}} M(\lambda) = \left(1 + Y \frac{1}{1 - X - Y} X\right) \frac{1}{1 - X - Y} = \sum_{k=0}^{\infty} (X+Y)^k$. The paper provides explicit matrix constructions and closed forms for restricted partition classes, including $\mathcal{OP}$, $\mathcal{SP}$, and $p$-core sets, suggesting potential generalizations (e.g., to higher-dimensional partitions) and offering practical tools for analyzing partition restrictions within a representation-theoretic context.

Abstract

In this paper, we introduce the generating functions of partition sequences. Partition sequences have a one-to-one correspondence with partitions. Therefore, the generating function has no multiplicity and appears meaningless initially. However, we show that using a matrix can give meaning to the coefficients and preserve valuable information about partitions. We also introduce some restrictions on partitions suitable for these generating functions.

Theorems & Definitions (16)

• Example 1
• Example 2
• Example 3
• Example 4
• Example 5
• Example 6
• Theorem 7
• proof
• Theorem 8
• proof