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On Weighted and Bounded Multidimensional Catalan Numbers

Ryota Inagaki, Dimana Pramatarova

TL;DR

This work introduces a weighted framework for multidimensional Catalan numbers via Balanced ballot paths in $k$ dimensions, using a height statistic to define bounded and weighted variants. It develops matrix-based recurrences for weighted 2D Catalan numbers and extends these ideas to higher dimensions, establishing periodicity criteria modulo an integer through finite-state transitions. The authors construct two new families of arrays: the $k$-dimensional Balanced-ballot-path-height triangles and a multidimensional Narayana triangle, providing recursive formulas, explicit counts, and tables for low-dimensional cases, along with data-driven observations and code to compute them. The results offer both theoretical insight into the structure of multidimensional Catalan-type enumerations and practical tools for exploring height- and peak-based statistics in higher-dimensional lattice paths, with potential connections to Standard Young Tableaux and related combinatorial objects.

Abstract

We define a weighted analog for the multidimensional Catalan numbers, obtain matrix-based recurrences for some of them, and give conditions under which they are periodic. Building on this framework, we introduce two new sequences of triangular arrays: the first one enumerates the $k$-dimensional Balanced ballot paths of exact height $s$; the second one is a new multidimensional generalization of the Narayana numbers, which count the number of Balanced ballot paths with exactly $p$ peaks.

On Weighted and Bounded Multidimensional Catalan Numbers

TL;DR

This work introduces a weighted framework for multidimensional Catalan numbers via Balanced ballot paths in dimensions, using a height statistic to define bounded and weighted variants. It develops matrix-based recurrences for weighted 2D Catalan numbers and extends these ideas to higher dimensions, establishing periodicity criteria modulo an integer through finite-state transitions. The authors construct two new families of arrays: the -dimensional Balanced-ballot-path-height triangles and a multidimensional Narayana triangle, providing recursive formulas, explicit counts, and tables for low-dimensional cases, along with data-driven observations and code to compute them. The results offer both theoretical insight into the structure of multidimensional Catalan-type enumerations and practical tools for exploring height- and peak-based statistics in higher-dimensional lattice paths, with potential connections to Standard Young Tableaux and related combinatorial objects.

Abstract

We define a weighted analog for the multidimensional Catalan numbers, obtain matrix-based recurrences for some of them, and give conditions under which they are periodic. Building on this framework, we introduce two new sequences of triangular arrays: the first one enumerates the -dimensional Balanced ballot paths of exact height ; the second one is a new multidimensional generalization of the Narayana numbers, which count the number of Balanced ballot paths with exactly peaks.
Paper Structure (10 sections, 17 theorems, 22 equations, 5 figures, 4 tables)

This paper contains 10 sections, 17 theorems, 22 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Definitions and Notations
  3. Problem Setup
  4. Prior and Basic Results on Weighted 2-Catalan Numbers
  5. Periodicity of multidimensional Catalan numbers
  6. Examples of Weighted $k$-dimensional Catalan Numbers
  7. Recursive formula for certain higher-dimensional weighted and bounded Catalan numbers
  8. Multidimensional Balanced-Ballot-Path-Height triangles
  9. A multidimensional generalization of the Narayana triangle
  10. Acknowledgments

Key Result

Lemma 2.12

The $2$-dimensional weighted Catalan numbers satisfy the following recurrence:

Figures (5)

  • Figure 1: A Dyck path of $8$ steps
  • Figure 2: A weighted Dyck path with $wt_{\vec{b}}(P) = b_0^2b_1^2$
  • Figure 3: For weight vector $\vec{b}= (b_0,b_1,b_2)$, all $5$ weighted Dyck paths of $6$ steps with corresponding weights for weight vector $\vec{b} = (b_0,b_1, b_2, \dots)$. The third weighted Catalan number for this weight vector is $C^{\vec{b}}_3 = b_0^3 + 2b_0^2b_1 + b_0b_1^2+ b_0b_1b_2$.
  • Figure 4: A 3-dimensional Balanced ballot path from $(0,0,0)$ to $(3,3,3)$ with the heights of each point along the path indicated. We use the formula $h_3(x) = x_1 - x_2 + x_1 - x_3$ to calculate the heights.
  • Figure 5: The possible states for $C^{\vec{b},3}_n$

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: ($A060854$ in OEIS oeis)
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 34 more