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

Yuxuan Zhang, Yan Zhuang

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.

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

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 -box paths, which are in bijection with -tuples of -ary trees. A bijection is given between -box paths and a subfamily of -Dyck paths, as well as a bijection with a subfamily of -threshold sequences. We also study the refined enumeration of -box paths by the number of returns and the number of long ascents. Notably, the distribution of long ascents over -box paths generalizes the Narayana distribution on Dyck paths, and we find that -box paths with exactly two long ascents provide a combinatorial model for the second -gonal numbers.
Paper Structure (7 sections, 13 theorems, 54 equations, 6 figures, 4 tables)

This paper contains 7 sections, 13 theorems, 54 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries: k-ary trees and k-Dyck paths
  3. k-box paths and their enumeration
  4. A bijection between k-box paths and (k+1)k-Dyck paths
  5. A bijection between k-box paths and (k+2)k-threshold sequences
  6. Counting k-box paths by returns
  7. Counting k-box paths by long ascents

Key Result

Proposition 3.1

Among skew Dyck paths with exactly $n$$UD^{k}L$-factors, the paths of smallest semilength are of semilength $(k+2)n-1$.

Figures (6)

  • Figure 1: The skew Dyck path corresponding to the word $UDU^{5}DL^{2}DUDL$
  • Figure 2: All box paths of size 3
  • Figure 3: A ternary tree with 11 nodes
  • Figure 4: A $3$-Dyck path and the corresponding augmented $3$-Dyck path
  • Figure 5: A $2$-box path and the corresponding $3_{2}$-Dyck path
  • ...and 1 more figures

Theorems & Definitions (28)

  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • Remark 3.6
  • ...and 18 more