10 problems

Sun's sequence of polynomials is defined by … A polynomial sequence $\{f_n(q)\}_{n\geq 0}$ is q-log-convex if, for every $n\geq 1$, the polynomial $f_{n+1}(q)f_{n-1}(q)-f_n^2(q)$ h…

For each $n\geq 1$, let $P_n(x)=\sum_k P_{n,k}x^k$, where $P_{n,k}$ is the number of permutations of $[n]=\{1,2,\ldots,n\}$ whose longest increasing subsequence has length $k$. A p…

Consider the triangular array with entries $\binom{n}{k}^2$ for $0\leq k\leq n$. Conditions (C1) and (C2) are the two conditions imposed on a triangular array in the source's theor…

The q-log-convexity conjecture. (i) The sequence $(\sum_{k=0}^n\binom nk^3q^k)_{n\geq8}$ is $q$-log-convex; for every integer $m\geq2$, the sequence…

Let $\{f_i(q)\}_{i\geq 0}$ be any of the polynomial sequences defined in Example $\mathrm{basic\text{-}qSM}$. A sequence is infinitely $q$-log-convex if every iterate under … has c…

For positive integers $r$ and $s$, define the generalized Apéry polynomials by … A polynomial sequence $(f_n(q))_{n\ge 0}$ is $q$-log-convex when the coefficients of…

Lin–Zeng's conjecture. The Jacobi–Stirling transformations of the two kinds preserve log-convexity for $z=0,1$. This conjecture concerns preservation of log-convexity by the transf…

Liu–Wang's conjecture. For every fixed $n$ and $r$, there is an integer $k'=k'(n,r)$ such that $\alpha(n,r,k)\geq 0$ for $k\leq k'$ and $\alpha(n,r,k)\leq 0$ for $k>k'$. Consequent…

For each $n\geq 1$, let $M_{2n}(x)=\sum_k M_{2n,k}x^k$, where $M_{2n,k}$ is the number of matchings on $[2n]$ with crossing number $k$. A polynomial sequence is infinitely q-log-co…

For each $n\geq 1$, let $P_n(x)=\sum_k P_{n,k}x^k$, where $P_{n,k}$ is the number of permutations of $[n]=\{1,2,\ldots,n\}$ whose longest increasing subsequence has length $k$. A p…