2014/11/10 by Ricky X. F. Chen, Chen, Ricky X. F., Christian M. Reidys +1 · 2 citations
Mathematics · Physics and Astronomy · #05A19 #05C05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1411.2530
openalex publication_date 2014/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note, by counting some colored plane trees we obtain several binomial identities. These identities can be viewed as specific evaluations of certain generalizations of the Narayana polynomials. As consequences, it provides combinatorial proofs for a bijective problem in Stanley's collection "Bijective Proof Problems", a new formula for the Narayana polynomials as well as a new expression for the Harer-Zagier formula enumerating unicellular maps, in a unified way. Furthermore, we identify a class of plane trees, whose enumeration is closely connected to the Schröder numbers. Many other binomial identities are presented as well.