On Triangle of Narayana Numbers
DOI:
https://doi.org/10.5281/zenodo.21899476Keywords:
Narayana numbers, Narayana triangle, Dyck paths, Catalan numbers, enumerative combinatorics, noncrossing partitions, rooted trees, 132-avoiding permutationsAbstract
This paper investigates the Triangle of Narayana Numbers as a refined combinatorial structure arising from Dyck paths classified according to the number of peaks. Using a theoretical-deductive and expository approach, the study establishes the definition and closed-form expression of the Narayana numbers, given by
where counts the number of Dyck paths of semilength with exactly peaks. The paper further examines the major algebraic and structural properties of the Narayana triangle, including boundary values, symmetry, recurrence relations, determinant and quadratic identities, central column formulas, parity patterns, operator representations, partial sums, and generating functions. In addition, it explores the relationship of the Narayana numbers with Catalan numbers, Pascal’s triangle, and unsigned Lah numbers, showing how these connections enrich their role in enumerative combinatorics. Selected applications are also discussed, particularly in noncrossing set partitions, unlabeled ordered rooted trees, and 132-avoiding permutations classified by descents. The findings affirm that the Narayana triangle is not merely a refinement of Catalan counting but a mathematically significant structure with broad combinatorial interpretations. Overall, the study contributes to the systematic presentation and deeper understanding of Narayana numbers as an important object in discrete mathematics and enumerative combinatorics.
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