On Schreier-type Sets, Partitions, and Compositions

TL;DR

This paper explores the relationship between sparse Schreier-type sets and partition numbers, establishing a bijection between certain sparse, strong Schreier sets and partitions with restricted parts, also extending results to compositions.

Contribution

It introduces a novel connection between sparse Schreier-type sets and restricted partition numbers, providing new combinatorial identities and extending the analysis to compositions.

Findings

The cardinality of certain sparse Schreier sets equals the number of partitions with restricted parts.

A bijection is established between sparse Schreier sets and partitions avoiding small parts.

Results extend to integer compositions, broadening the combinatorial framework.

Abstract

A nonempty set $A\subset\mathbb{N}$ is $\ell$-strong Schreier if $\min A\geqslant \ell|A|-\ell+1$. We define a set of positive integers to be sparse if either the set has at most two numbers or the differences between consecutive numbers in increasing order are non-decreasing. This note establishes a connection between sparse Schreier-type sets and (restricted) partition numbers. One of our results states that if $\mathcal{G}_{n,\ell}$ consists of partitions of $n$ that contain no parts in $\{2, \ldots, \ell\}$, and \begin{equation*} \mathcal{A}_{n,\ell} \ :=\ \{A\subset \{1, \ldots, n\}\,:\, n\in A, A\mbox{ is sparse and }\ell\mbox{-strong Schreier}\}, \end{equation*} then $$|\mathcal{A}_{n,\ell}|\ =\ |\mathcal{G}_{n-1,\ell}|, \quad n, \ell\in \mathbb{N}.$$ The special case $\mathcal{G}_{n-1, 1}$ consists of all partitions of $n-1$. Besides partitions, integer compositions are also investigated.