On Connection between the Numbers of Permutations and Full Cycles with Some Restrictions on Positions and Up-Down Structure

TL;DR

This paper explores the relationships between the counts of permutations and full cycles, especially under restrictions like specific position constraints and up-down patterns, revealing new combinatorial connections.

Contribution

It introduces novel links between permutation counts and full cycles with restrictions, focusing on up-down structures and position constraints.

Findings

Identifies new combinatorial relationships

Establishes formulas connecting permutations and cycles with restrictions

Provides insights into structured permutation enumeration

Abstract

We discuss both simple and more subtle connections between the numbers of permutations and full cycles with some restrictions,in particular, between the numbers of permutations and full cycles with prescribed up-down structure.