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Let $P_{n,k}$ be the number of permutations $\pi$ on [n]={1, 2,..., n} suchthat the length of the longest increasing subsequences of $\pi$ equals k, andlet $M_{2n, k}$ be the number of matchings on [2n] with crossing number k.Define $P_n(x)= \sum_k P_{n,k}x^k$ and $M_{2n}(x)=\sum_{k} M_{2n,k}x^k$. Wepropose some conjectures on the log-concavity and q-log-convexity of thepolynomials $P_n(x)$ and $M_{2n}(x)$. We also introduce the notions of$\infty$-q-log-convexity and $\infty$-q-log-concavity, and the notion of higherorder log-concavity with respect to $\infty$-q-log-convex or$\infty$-q-log-concavity. A conjecture on the $\infty$-q-log-convexity of theBoros-Moll polynomials is presented. It seems that $M_{2n}(x)$ are log-concaveof any order with respect to $\infty$-q-log-convexity.

Log-concavity and q-Log-convexity Conjectures on the Longest Increasing Subsequences of Permutations