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Further results on staircase graph words

Sela Fried

TL;DR

This paper generalizes staircase words to staircase graph words on the distance-$L$ path $P_{n,L}$, counting words of length $n$ over an alphabet $[k]$ with the constraint $\max\{w_i,\dots,w_{i+L}\}-\min\{w_i,\dots,w_{i+L}\}\le 1$. Using the kernel method, the authors derive a closed-form generating function $f(x)=f_{k,L}(x)$, expressing it through Chebyshev polynomials of the second kind $U_k(\phi)$ with $\phi=\left(1-\frac{x^{2L}-x}{1-x}\right)/(2x^{L})$ and auxiliary polynomials. The framework unifies and extends earlier $L=1$ results (Knopfmacher et al.) and provides explicit forms for small $L$ (e.g., $L=1,2$) and several $k$, enabling exact enumeration and potential asymptotic analysis. These results enhance understanding of restricted-word enumerations on distance-$L$ path graphs and highlight the role of Chebyshev polynomials in combinatorial generating functions.

Abstract

Staircase words are words in which consecutive letters do not differ by more than $1$. We generalize this by extending the restriction to letters lying further apart from each other and obtain the corresponding generating functions, which we express in terms of the Chebyshev polynomials of the second kind.

Further results on staircase graph words

TL;DR

This paper generalizes staircase words to staircase graph words on the distance- path , counting words of length over an alphabet with the constraint . Using the kernel method, the authors derive a closed-form generating function , expressing it through Chebyshev polynomials of the second kind with and auxiliary polynomials. The framework unifies and extends earlier results (Knopfmacher et al.) and provides explicit forms for small (e.g., ) and several , enabling exact enumeration and potential asymptotic analysis. These results enhance understanding of restricted-word enumerations on distance- path graphs and highlight the role of Chebyshev polynomials in combinatorial generating functions.

Abstract

Staircase words are words in which consecutive letters do not differ by more than . We generalize this by extending the restriction to letters lying further apart from each other and obtain the corresponding generating functions, which we express in terms of the Chebyshev polynomials of the second kind.

Paper Structure

This paper contains 3 sections, 3 theorems, 19 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Lemma 1

Let $I=\{0,1\}^L\setminus\{(0,\ldots,0),(1,\ldots,1)\}, F_{0,\ldots,0}(x,t)=\sum_{a=1}^{k}f_{a,\ldots,a}(x)t^{a-1}$, and, for each $(o_1,\ldots,o_L)\in I$, let $F_{o_1,\ldots,o_L}(x,t)=\sum_{a=1}^{k-1}f_{a+o_1,\ldots,a+o_L}(x)t^{a-1}$. Then

Figures (1)

  • Figure 1: The graph $P_{6,2}$

Theorems & Definitions (7)

  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 3
  • proof
  • Example 4
  • Example 5