An analogue of the Erd{\H o}s Matching Conjecture for permutations with fixed number of cycles

TL;DR

This paper investigates the maximum size of permutation families with a fixed number of cycles, under constraints on the size of matchings, extending combinatorial principles similar to the Erdős Matching Conjecture.

Contribution

It introduces a new bound for the size of permutation families with limited matchings, generalizing classical combinatorial results to permutations with fixed cycle counts.

Findings

Derived maximum size bounds for permutation families with constrained matchings.

Extended Erdős Matching Conjecture concepts to permutation sets with fixed cycle numbers.

Provided structural insights into permutation families under matching restrictions.

Abstract

Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For each integer $k\geq 1$, let $S_{n,k}$ be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles. A subset $H\subseteq S_{n,k}$ is to be a matching if $\pi_1$ and $\pi_2$ do not have any common cycles for all distinct $\pi_1,\pi_2\in H$. The matching number of a family $\mathcal A\subseteq S_{n,k}$ is denoted by $\nu_{p}(\mathcal A)$ and is defined to be the size of the largest matching in $\mathcal A$. In this paper, we determine the maximum size of a family $\mathcal A\subseteq S_{n,k}$ subject to the condition $\nu_p(\mathcal A)\leq s$.