Statistics on bargraphs viewed as cornerless Motzkin paths

TL;DR

This paper establishes a bijection between bargraphs and cornerless Motzkin paths, enabling new enumerations and analyses of bargraphs, including symmetric and alternating variants, with simpler derivations and novel results.

Contribution

It introduces a bijection between bargraphs and cornerless Motzkin paths, simplifying enumeration and analysis of various bargraph subclasses.

Findings

Enumerated bargraphs with respect to multiple statistics.

Derived simpler formulas for known results.

Counted symmetric and alternating bargraphs.

Abstract

A bargraph is a self-avoiding lattice path with steps $U=(0,1)$, $H=(1,0)$ and $D=(0,-1)$ that starts at the origin and ends on the $x$-axis, and stays strictly above the $x$-axis everywhere except at the endpoints. Bargraphs have been studied as a special class of convex polyominoes, and enumerated using the so-called wasp-waist decomposition of Bousquet-M\'elou and Rechnitzer. In this paper we note that there is a trivial bijection between bargraphs and Motzkin paths without peaks or valleys. This allows us to use the recursive structure of Motzkin paths to enumerate bargraphs with respect to several statistics, finding simpler derivations of known results and obtaining many new ones. We also count symmetric bargraphs and alternating bargraphs. In some cases we construct statistic-preserving bijections between different combinatorial objects, proving some identities that we encounter along the way.