A subfamily of skew Dyck paths related to $k$-ary trees

TL;DR

This paper introduces a new class of skew Dyck paths called box paths, establishes their bijections with various combinatorial structures, and explores their enumeration and distribution properties, generalizing known distributions and connecting to polygonal numbers.

Contribution

It defines k-box paths and their bijections with trees and other structures, extending the combinatorial understanding of skew Dyck paths and related sequences.

Findings

k-box paths are in bijection with (k+1)-tuples of (k+2)-ary trees

Distribution of long ascents generalizes Narayana distribution

Certain k-box paths model second k-gonal numbers

Abstract

We introduce a subfamily of skew Dyck paths called box paths and show that they are in bijection with pairs of ternary trees, confirming an observation stated previously on the On-Line Encyclopedia of Integer Sequences. More generally, we define $k$-box paths, which are in bijection with $(k+1)$-tuples of $(k+2)$-ary trees. A bijection is given between $k$-box paths and a subfamily of $k_{t}$-Dyck paths, as well as a bijection with a subfamily of $(k,\ell)$-threshold sequences. We also study the refined enumeration of $k$-box paths by the number of returns and the number of long ascents. Notably, the distribution of long ascents over $k$-box paths generalizes the Narayana distribution on Dyck paths, and we find that $(k-3)$-box paths with exactly two long ascents provide a combinatorial model for the second $k$-gonal numbers.