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Counting words without strictly increasing subwords of fixed length

Senan Sekhon

TL;DR

This work develops exact generating-function formulas for counting $n$-ary words that avoid strictly increasing subwords of fixed length $k$, employing the Goulden-Jackson cluster method and root-of-unity techniques. It provides closed-form expressions for the streak-avoiding generating function and extends the analysis to soft streaks via conjectures, yielding related expressions and special cases. The authors connect these counts to random-sampling metrics, giving explicit formulas for waiting-times and exploring the continuous limit as the alphabet size grows, with asymptotic behavior and radius-of-convergence properties discussed. The results offer exact enumeration tools, asymptotics, and practical implications for sampling and combinatorial analysis of monotone-subword avoidance.

Abstract

In this paper, we derive exact formulas for generating functions counting the number of $n$-ary words avoiding strictly increasing subwords of length $k$, and provide some applications of these formulas.

Counting words without strictly increasing subwords of fixed length

TL;DR

This work develops exact generating-function formulas for counting -ary words that avoid strictly increasing subwords of fixed length , employing the Goulden-Jackson cluster method and root-of-unity techniques. It provides closed-form expressions for the streak-avoiding generating function and extends the analysis to soft streaks via conjectures, yielding related expressions and special cases. The authors connect these counts to random-sampling metrics, giving explicit formulas for waiting-times and exploring the continuous limit as the alphabet size grows, with asymptotic behavior and radius-of-convergence properties discussed. The results offer exact enumeration tools, asymptotics, and practical implications for sampling and combinatorial analysis of monotone-subword avoidance.

Abstract

In this paper, we derive exact formulas for generating functions counting the number of -ary words avoiding strictly increasing subwords of length , and provide some applications of these formulas.

Paper Structure

This paper contains 8 sections, 20 theorems, 77 equations.

Table of Contents

  1. Preliminary Definitions
  2. The Goulden-Jackson Cluster Method
  3. Radius of convergence
  4. Avoiding streaks of length $k$
  5. Avoiding soft streaks of length $k$
  6. Application to Random Sampling
  7. Continuous limit
  8. Lower bound for the radius of convergence

Key Result

Lemma 1.2

$\psi_{k,r}$ can be expressed as follows: In other words, $\psi_{k,r}$ is the average of the values of the function $z^r\qty(1-\frac{1}{z})$ as $z$ ranges over all the $k$th roots of unity. The term with $s=0$ can be omitted as this is zero.

Theorems & Definitions (43)

  • Definition 1.1
  • Remark
  • Lemma 1.2
  • Remark
  • Proposition 1.3
  • Corollary 1.4
  • Lemma 1.5
  • Definition 2.1
  • Example
  • Example
  • ...and 33 more