Bijections between generalized Catalan families of types A and C
Myrto Kallipoliti, Eleni Tzanaki
TL;DR
This paper explores the relationship between generalized Catalan numbers of types A and C, establishing bijections between their combinatorial structures and regions of associated hyperplane arrangements.
Contribution
It introduces a novel bijection linking m-Dyck paths and NE lattice paths within rectangles, connecting type A and C Catalan combinatorics.
Findings
Established a bijection between m-Dyck paths and NE lattice paths.
Connected dominant regions of m-Shi arrangements for types A and C.
Extended Catalan number relations to combinatorial structures.
Abstract
Motivated by the relation holding for the m-generalized Catalan numbers of type A and C, the connection between dominant regions of the m-Shi arrangement of type A and C is investigated. In the same line of thought, a bijection between mn+1 copies of each m-Dyck path of height n and the set of NE lattice paths inside an n times mn rectangle is provided.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Advanced Algebra and Geometry