Enumeration in the lattice of $q$-decreasing words

2025-11-12

AI Summary

This paper investigates the poset structure of $q$-decreasing words under the componentwise order. **Problem:** Characterizing the lattice-theoretic and combinatorial properties of this poset for arbitrary positive rational $q$. **Method:** Combining lattice theory, combinatorial enumeration, pattern-avoidance analysis, and asymptotic techniques—employing both algebraic derivations and explicit counting arguments. **Contribution/Results:** First, we prove that the set of $q$-decreasing words forms a lattice for every positive rational $q$, establishing its lattice structure for the first time. Second, we derive closed-form formulas for the number of join-irreducible elements; fully characterize covering relations, interval counts, and the number of meet-irreducible elements; and establish a bijection between meet-irreducibles and words over a finite alphabet avoiding length-2 consecutive patterns. These results deepen the algebraic understanding of $q$-decreasing words and forge novel connections to restricted word enumeration, yielding precise asymptotic characterizations of several structural parameters.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Natural Language Processing: Lexical Semantics and Morphology
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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 $2lceil q ceil+1$ letters avoiding $lceil q ceil^2+2lceil q ceil-1$ consecutive patterns of length 2. Furthermore, we analyze the asymptotic behavior of several of these quantities.

Problem

• Proves the poset of q-decreasing words forms a lattice structure
• Enumerates join-irreducible elements and counts coverings and intervals
• Analyzes asymptotic behavior of quantities in q-decreasing word lattices

Innovation

• Proves poset of q-decreasing words forms lattice
• Enumerates join-irreducible elements for arbitrary q
• Determines coverings intervals meet-irreducible elements structure