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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.
In this paper, we define four transformations on the classical Catalan triangle $mathcal{C}=(C_{n,k})_{ngeq kgeq 0}$ with $C_{n,k}=frac{k+1}{n+1}binom{2n-k}{n}$. The first three ones are based on the determinant and the forth is utilizing the permane
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 diffe
A polynomial $A(q)=sum_{i=0}^n a_iq^i$ is said to be unimodal if $a_0le a_1le cdots le a_kge a_{k+1} ge cdots ge a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)= frac{1}{[n+m]} left[ m+n at
The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of $n$. This pape
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