Some Elementary Partition Inequalities and Their Implications

TL;DR

This paper establishes new inequalities between partition counts with specific restrictions on largest and smallest parts, revealing interesting combinatorial relationships and implications in partition theory.

Contribution

It introduces novel inequalities between partition counts with bounds on largest parts and restrictions on part occurrences, expanding understanding of partition combinatorics.

Findings

Number of partitions with smallest part 1 exceeds those with smallest part greater than 1 under certain bounds.

Partition inequalities imply various combinatorial consequences.

Results deepen the understanding of partition restrictions and their relationships.

Abstract

We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for $L\geq 1$, the number of partitions with $l-s \leq L$ and $s=1$ is greater than the number of partitions with $l-s\leq L$ and $s>1$. Here $l$ and $s$ are the largest part and the smallest part of the partition, respectively.