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We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities.
In this note, we present several identities involving binomial coefficients and the two kind of Stirling numbers.
We give a counting formula for the set of rectangular increasing tableaux in terms of generalized Narayana numbers. We define small $m$-Schroder paths and give a bijection between the set of increasing rectangular tableaux and small $m$-Schroder path
Given a permutation $f$, we study the positroid Catalan number $C_f$ defined to be the torus-equivariant Euler characteristic of the associated open positroid variety. We introduce a class of repetition-free permutations and show that the correspondi
We consider Tuenter polynomials as linear combinations of descending factorials and show that coefficients of these linear combinations are expressed via a Catalan triangle of numbers. We also describe a triangle of coefficients in terms of some polynomials.
The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the nth Catalan number. In this paper, we will generalize this setup to