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Enumeration in the lattice of $q$-decreasing words

Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov

TL;DR

This work analyzes the lattice structure of $q$-decreasing words under componentwise order, proving that the poset $\mathbb{W}_n^q$ is a lattice for every $q>0$ and providing detailed enumerations of join-irreducible and meet-irreducible elements, coverings, and intervals. The authors develop generating-function techniques, floor/ceiling identities, and a pattern-avoidance translation to handle rational and irrational $q$, yielding closed forms for rational $q$ and effective computation methods otherwise; the asymptotics are tied to the function $\Phi(q)$. For $0<q\le1$ meet-irreducibles decompose into blocks $0^a1^b$ and are characterized by avoiding a $\mathtt{CB}$-pattern, while for $q>1$ the meet-irreducibles correspond to pattern-avoiding words on an expanded alphabet of size $2\lceil q\rceil+1$, avoiding a large set of length-2 patterns; these analyses give explicit generating functions and illuminate connections to pattern-avoidance theory and generalized Fibonacci-type growth. Overall, the paper provides exact and asymptotic counts for key lattice parameters, linking $q$-decreasing words to pattern-avoidance and offering a framework for further combinatorial enumeration of related word-structures.

Abstract

We prove that the poset of $q$-decreasing words equipped with the componentwise order forms a lattice. We enumerate the join-irreducible elements for arbitrary $q>0$, and for any positive rational number $q$, we determine the number of coverings, intervals and meet-irreducible elements. The latter present the same structure as words over an alphabet of $2\lceil q\rceil+1$ letters avoiding $\lceil q\rceil^2+2\lceil q\rceil-1$ consecutive patterns of length 2. Furthermore, we analyze the asymptotic behavior of several of these quantities.

Enumeration in the lattice of $q$-decreasing words

TL;DR

This work analyzes the lattice structure of -decreasing words under componentwise order, proving that the poset is a lattice for every and providing detailed enumerations of join-irreducible and meet-irreducible elements, coverings, and intervals. The authors develop generating-function techniques, floor/ceiling identities, and a pattern-avoidance translation to handle rational and irrational , yielding closed forms for rational and effective computation methods otherwise; the asymptotics are tied to the function . For meet-irreducibles decompose into blocks and are characterized by avoiding a -pattern, while for the meet-irreducibles correspond to pattern-avoiding words on an expanded alphabet of size , avoiding a large set of length-2 patterns; these analyses give explicit generating functions and illuminate connections to pattern-avoidance theory and generalized Fibonacci-type growth. Overall, the paper provides exact and asymptotic counts for key lattice parameters, linking -decreasing words to pattern-avoidance and offering a framework for further combinatorial enumeration of related word-structures.

Abstract

We prove that the poset of -decreasing words equipped with the componentwise order forms a lattice. We enumerate the join-irreducible elements for arbitrary , and for any positive rational number , we determine the number of coverings, intervals and meet-irreducible elements. The latter present the same structure as words over an alphabet of letters avoiding consecutive patterns of length 2. Furthermore, we analyze the asymptotic behavior of several of these quantities.

Paper Structure

This paper contains 10 sections, 16 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Useful results
  3. Join-irreducible elements and coverings
  4. Join-irreducible elements
  5. Coverings
  6. Intervals
  7. Meet-irreducible elements in $\mathbb{W}_n^q$ for rational numbers $q>0$
  8. Meet-irreducible elements in $\mathbb{W}_n^q$ for a rational number $0<q\leq 1$
  9. Meet-irreducible elements in $\mathbb{W}_n^q$ for a rational number $q>1$
  10. Acknowledgement

Key Result

Theorem 1.1

For $q\geq 0$, the poset $\mathbb{W}_n^q$ is a lattice for any $n\geq 1$.

Figures (1)

  • Figure 1: The lattice $\mathbb{W}_5^1$. It contains 20 coverings (edges), 7 meet-irreducible elements, 5 join-irreducible elements and 56 intervals.

Theorems & Definitions (41)

  • Theorem 1.1
  • proof
  • proof
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • Lemma 2.6
  • ...and 31 more