Combinatorics of Triangular Partitions

TL;DR

This paper explores the combinatorics of triangular partitions, generalizing classical concepts like Dyck paths and parking functions, and reveals that the Young lattice restricted to these partitions has a planar Hasse diagram.

Contribution

It introduces the combinatorics of triangular partitions and demonstrates their properties, including a planar Young lattice and generalized enumeration methods.

Findings

Young lattice restricted to triangular partitions is planar

Generalization of Dyck path enumeration to triangular partitions

Triangular partitions include classical combinatorial objects as special cases

Abstract

The aim of this paper is to develop the combinatorics of constructions associated to what we call \emph{triangular partitions}. As introduced in arXiv:2102.07931, these are the partitions whose cells are those lying below the line joining points $(r,0)$ and $(0,s)$, for any given positive reals $r$ and $s$. Classical notions such as Dyck paths and parking functions are naturally generalized by considering the set of partitions included in a given triangular partition. One of our striking results is that the restriction of the Young lattice to triangular partition has a planar Hasse diagram, with many nice properties. It follows that we may generalize the "first-return" recurrence, for the enumeration of classical Dyck paths, to the enumeration of all partitions contained in a fixed triangular one.