Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Log-concavity and q-Log-convexity Conjectures on the Longest Increasing
  Subsequences of Permutations   Let $P_{n,k}$ be the number of permutations $\pi$ on [n]={1, 2,..., n} such
that the length of the longest increasing subsequences of $\pi$ equals k, and
let $M_{2n, k}$ be the number of matchings on [2n] with crossing number k.
Define $P_n(x)= \sum_k P_{n,k}x^k$ and $M_{2n}(x)=\sum_{k} M_{2n,k}x^k$. We
propose some conjectures on the log-concavity and q-log-convexity of the
polynomials $P_n(x)$ and $M_{2n}(x)$. We also introduce the notions of
$\infty$-q-log-convexity and $\infty$-q-log-concavity, and the notion of higher
order log-concavity with respect to $\infty$-q-log-convex or
$\infty$-q-log-concavity. A conjecture on the $\infty$-q-log-convexity of the
Boros-Moll polynomials is presented. It seems that $M_{2n}(x)$ are log-concave
of any order with respect to $\infty$-q-log-convexity.

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The categories of this paper are: comp-gas
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int