Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Context-free Grammars for Permutations and Increasing Trees   In this paper, we introduce the notion of a grammatical labeling to describe
a recursive process of generating combinatorial objects based on a context-free
grammar. For example, by labeling the ascents and descents of a Stirling
permutation, we obtain a grammar for the second-order Eulerian polynomials. By
using the grammar for $0$-$1$-$2$ increasing trees given by Dumont, we obtain a
grammatical derivation of the generating function of the Andr\'e polynomials
obtained by Foata and Sch\"utzenberger, without solving a differential
equation. We also find a grammar for the number $T(n,k)$ of permutations of
$[n]=\{1,2,\ldots, n\}$ with $k$ exterior peaks, which was independently
discovered by Ma. We demonstrate that Gessel's formula for the generating
function of $T(n,k)$ can be deduced from this grammar. Moreover, by using
grammars we show that the number of the permutations of $[n]$ with $k$ exterior
peaks equals the number of increasing trees on $[n]$ with $2k+1$ vertices of
even degree. A combinatorial proof of this fact is also presented.

6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th