# A bijection between bargraphs and Dyck paths

## TL;DR

This paper introduces a novel bijection between bargraphs and Dyck paths, providing new insights into their combinatorial properties and a new interpretation of Catalan numbers related to bargraph statistics.

## Contribution

It presents an unusual bijection between bargraphs and Dyck paths and explores how various statistics are preserved or transformed, offering new combinatorial interpretations.

## Key findings

- New bijection between bargraphs and Dyck paths
- Catalan numbers interpreted via bargraph statistics
- Mapping of statistics between the two structures

## Abstract

Bargraphs are a special class of convex polyominoes. They can be identified with lattice paths with unit steps north, east, and south that start at the origin, end on the $x$-axis, and stay strictly above the $x$-axis everywhere except at the endpoints. Bargraphs, which are used to represent histograms and to model polymers in statistical physics, have been enumerated in the literature by semiperimeter and by several other statistics, using different methods such as the wasp-waist decomposition of Bousquet-M\'elou and Rechnitzer, and a bijection with certain Motzkin paths. In this paper we describe an unusual bijection between bargraphs and Dyck paths, and study how some statistics are mapped by the bijection. As a consequence, we obtain a new interpretation of Catalan numbers, as counting bargraphs where the semiperimeter minus the number of peaks is fixed.