Lattice Path Enumeration and Its Applications in Representation Theory
Jianqiang Feng, Wenli Liu, Ximei Bai, Zhenheng Li
TL;DR
This paper explores lattice path enumeration under constraints and applies these results to compute module dimensions in representation theory, revealing connections to Catalan numbers and deriving combinatorial identities.
Contribution
It introduces new enumeration formulas for constrained lattice paths and applies them to determine module dimensions, linking combinatorics with algebraic representation theory.
Findings
Catalan numbers appear as module dimensions
Derived new combinatorial identities
Established formulas for submodule dimensions
Abstract
In this paper, we enumerate lattice paths with certain constraints and apply the corresponding results to develop formulas for calculating the dimensions of submodules of a class of modules for planar upper triangular rook monoids. In particular, we show that the famous Catalan numbers appear as the dimensions of some special modules; we also obtain some combinatorial identities
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Commutative Algebra and Its Applications · Algebraic structures and combinatorial models