Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Enumeration of m-ary cacti   The purpose of this paper is to enumerate various classes of cyclically
colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is
motivated by the topological classification of complex polynomials having at
most m critical values, studied by Zvonkin and others. We obtain explicit
formulae for both labelled and unlabelled m-ary cacti, according to i) the
number of polygons, ii) the vertex-color distribution, iii) the vertex-degree
distribution of each color. We also enumerate m-ary cacti according to the
order of their automorphism group. Using a generalization of Otter's formula,
we express the species of m-ary cacti in terms of rooted and of pointed cacti.
A variant of the m-dimensional Lagrange inversion is then used to enumerate
these structures. The method of Liskovets for the enumeration of unrooted
planar maps can also be adapted to m-ary cacti.

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The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
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The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
18
The categories of this paper are: econ
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The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat