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Register a new userOn Parity Unimodality of $q$-Catalan Polynomials
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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 $C_{m,n}(q)= frac{1}{[n+m]} left[ m+n atop nright]$ for a coprime pair of positive integers $(m,n)$. We conjecture that they are unimodal with respect to parity, or equivalently, $(1+q)C_{m+n}(q)$ is unimodal. By using generating functions and the constant term method, we verify our conjecture for $mle 5$ in a straightforward way.
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In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillovs unimodality conjecture with the help of Newtons inequality. By using an equidistribution property between descent numbers and ascent numbers on ballot paths due to Sulanke and a bijection between lattice words and standard Young tableaux, we show that the rectangular Narayana polynomial is equal to the descent generating function on standard Young tableaux of certain rectangular shape, up to a power of the indeterminate. Then we obtain the real-rootedness of the rectangular Narayana polynomial based on Brentis result that the descent generating function of standard Young tableaux has only real zeros.
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 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 paper proves the equivalence of the two definitions for all $mgeq 1$ and all $nleq 4$. We also give a bijective proof of the joint symmetry property $C^{(m)}_n(q,t)=C^{(m)}_n(t,q)$ for all $mgeq 1$ and all $nleq 4$. The proof is based on a general approach for proving joint symmetry that dissects a collection of objects into chains, and then passes from a joint symmetry property of initial points and terminal points to joint symmetry of the full set of objects. Further consequences include unimodality results and specific formulas for the coefficients in $C^{(m)}_n(q,t)$ for all $mgeq 1$ and all $nleq 4$. We give analogous results for certain rational-slope $q,t$-Catalan polynomials.
In this paper we shall survey the various methods of evaluating Hankel determinants and as an illustration we evaluate some Hankel determinants of a q-analogue of Catalan numbers. Here we consider $frac{(aq;q)_{n}}{(abq^{2};q)_{n}}$ as a q-analogue of Catalan numbers $C_{n}=frac1{n+1}binom{2n}{n}$, which is known as the moments of the little q-Jacobi polynomials. We also give several proofs of this q-analogue, in which we use lattice paths, the orthogonal polynomials, or the basic hypergeometric series. We also consider a q-analogue of Schroder Hankel determinants, and give a new proof of Moztkin Hankel determinants using an addition formula for ${}_2F_{1}$.
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.
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