Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bijective enumeration of some colored permutations given by the product
  of two long cycles   Let $\gamma_n$ be the permutation on $n$ symbols defined by $\gamma_n = (1\
2\...\ n)$. We are interested in an enumerative problem on colored
permutations, that is permutations $\beta$ of $n$ in which the numbers from 1
to $n$ are colored with $p$ colors such that two elements in a same cycle have
the same color. We show that the proportion of colored permutations such that
$\gamma_n \beta^{-1}$ is a long cycle is given by the very simple ratio
$\frac{1}{n- p+1}$. Our proof is bijective and uses combinatorial objects such
as partitioned hypermaps and thorn trees. This formula is actually equivalent
to the proportionality of the number of long cycles $\alpha$ such that
$\gamma_n\alpha$ has $m$ cycles and Stirling numbers of size $n+1$, an
unexpected connection previously found by several authors by means of algebraic
methods. Moreover, our bijection allows us to refine the latter result with the
cycle type of the permutations.

25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
8
The categories of this paper are: hep-ex
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
9
The categories of this paper are: nlin
3
The categories of this paper are: physics