Narayana polynomials and some generalizations

Christian M. Reidys, Ricky X. F. Chen

Publication date: 10 November 2014

Abstract: 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 Schr"oder numbers. Many other binomial identities are presented as well.

Mathematics Subject Classification ID

Trees (05C05) Combinatorial identities, bijective combinatorics (05A19)

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