Solution to a BCC 2022 problem

TL;DR

This paper establishes an explicit bijection between certain words with specific letter counts and subgraphs of a labeled cycle with a fixed number of edges and components, solving a problem posed at a 2022 conference.

Contribution

It introduces a novel explicit correspondence between combinatorial words and cycle subgraphs, addressing a problem from the 2022 British Combinatorial Conference.

Findings

Established a bijection between words and cycle subgraphs.

Solved a specific combinatorial problem from a recent conference.

Provides a new combinatorial interpretation for the problem.

Abstract

For positive integers $n$ and $k$ such that $k$ is at most $n$, we find an explicit one-to-one correspondence between the following two sets: the set of words consisting of $k$ $R$s, $k$ $U$s, and $n - k$ $D$s, where the first letter of the word is not $D$; and the set of subgraphs $H$ of a cycle of length $2n$ (where that cycle has differently labelled vertices) such that $H$ has $n$ edges and $k$ connected components. This solves a problem of Thomas Selig from the 29th British Combinatorial Conference held at Lancaster University in July 2022.