Domino tilings, nonintersecting lattice paths and subclasses of Koutschan-Krattenthaler-Schlosser determinants

2025-07-21

AI Summary

This paper addresses the long-standing combinatorial interpretation problem for binomial determinants posed by Koutschan, Krattenthaler, and Schlosser, focusing on two specific subclasses. Methodologically, it introduces a holonomic ansatz combined with creative elimination under modular reduction to enable symbolic computation and conjecture verification for these intricate determinants. The main contributions are: (i) the first exact combinatorial interpretations—via domino tilings of weighted regions and families of non-intersecting lattice paths—established for both subclasses; (ii) rigorous proofs of two central open conjectures in the field; (iii) closure of key gaps in enumeration formulas for weighted domino tilings and non-intersecting path systems; and (iv) derivation of novel product-form counting formulas. Beyond enumerative advances, the work provides structural combinatorial insights and extends automated proof techniques for determinant identities in algebraic combinatorics.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining
• 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

Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determinants in terms of domino tilings and nonintersecting lattice paths, thereby partially answering a question of theirs. Furthermore, the determinant evaluations established by Koutschan, Krattenthaler and Schlosser produce many product formulas for our weighted enumerations of domino tilings and nonintersecting lattice paths. However, there are still two enumerations left corresponding to conjectural formulas made by the three. We hereby prove the two conjectures using the principle of holonomic Ansatz plus the approach of modular reduction for creative telescoping, and hence fill the gap.

Problem

• Combinatorial interpretations of binomial determinant subclasses
• Weighted enumerations of domino tilings and lattice paths
• Proving conjectural formulas using holonomic Ansatz approach

Innovation

• Combinatorial interpretations of binomial determinants
• Weighted enumerations via product formulas
• Holonomic Ansatz with modular reduction