2025/11/17 by Yu Hin Au, Au, Yu Hin, Murray R. Bremner +1
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2511.13671
openalex publication_date 2025/11/17 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28
We introduce and study a generalization of the Narayana numbers Nd(n,k) = (1)/(n+1) \binomn+1k+1 \binom n + (n-k)(d-2)+1k for integers d ≥ 2 and n,k ≥ 0. This two-parameter array extends the classical Narayana numbers (d=2) and yields a d-ary analogue of the Catalan numbers Cd(n) = ∑k=0n Nd(n,k). We give nine combinatorial interpretations of Nd(n,k) that unify and generalize known combinatorial interpretations of the Narayana numbers and C3(n) in the literature. In particular, we show that Nd(n,k) counts a natural class of operator monomials over a d-ary associative algebra, thereby extending a result of Bremner and Elgendy for the binary case. We also construct explicit bijections between these monomials and several families of classic combinatorial objects, including Schröder paths, Dyck paths, rooted ordered trees, and 231-avoiding permutations.