Abstract
A polynomial $A(q)=\sum_{i=0}^n a_iq^i$ is said to be unimodal if $a_0\leqslant a_1\leqslant \cdots \leqslant a_k\geqslant a_{k+1} \geqslant \cdots \geqslant a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)= \frac{1}{[n+m]} {m+n \brack n}_q $ 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 our conjecture for $m\le 5$ in a straightforward way.
Highlights
We will consider the unimodality of some symmetric polynomials
We investigate the unimodality of rational q-Catalan polynomials, which is defined to be
We find that Conjecture 1.2 can be extended for rational q-Catalan polynomials
Summary
We will consider the unimodality of some symmetric polynomials. A sequence a0, . . . , an is said to be symmetric if ai = an−i for all i. A polynomial P (q) = a0 + a1q + · · · + anqn of degree n is said to be symmetric (resp., unimodal) if its coefficient sequence a0, a1, . The q-Catalan polynomials Cn(q) are unimodal with respect to parity. For a coprime pair of positive integers (m, n), the (m, n)-rational qCatalan polynomials Cm,n(q) are unimodal with respect to parity. For a pair of positive integers (m, n), the polynomial Cm,n(q) is unimodal with respect to parity. 1.3 includes Conjecture 1.2 as a special case, since Cn+1,n(q) = Cn(q) We state the latter separately because Cn(q) has a different combinatorial interpretation. We introduce the combinatorial interpretations of q-Catalan polynomials
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