Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Plane permutations and applications to a result of Zagier-Stanley and
  distances of permutations   In this paper, we introduce plane permutations, i.e. pairs
$\mathfrak{p}=(s,\pi)$ where $s$ is an $n$-cycle and $\pi$ is an arbitrary
permutation, represented as a two-row array. Accordingly a plane permutation
gives rise to three distinct permutations: the permutation induced by the upper
horizontal ($s$), the vertical $\pi$) and the diagonal ($D_{\mathfrak{p}}$) of
the array. The latter can also be viewed as the three permutations of a
hypermap. In particular, a map corresponds to a plane permutation, in which the
diagonal is a fixed point-free involution. We study the transposition action on
plane permutations obtained by permuting their diagonal-blocks. We establish
basic properties of plane permutations and study transpositions and exceedances
and derive various enumerative results. In particular, we prove a recurrence
for the number of plane permutations having a fixed diagonal and $k$ cycles in
the vertical, generalizing Chapuy's recursion for maps filtered by the genus.
As applications of this framework, we present a combinatorial proof of a result
of Zagier and Stanley, on the number of $n$-cycles $\omega$, for which the
product $\omega(1~2~\cdots ~n)$ has exactly $k$ cycles. Furthermore, we
integrate studies on the transposition and block-interchange distance of
permutations as well as the reversal distance of signed permutations. Plane
permutations allow us to generalize and recover various lower bounds for
transposition and block-interchange distances and to connect reversals with
block-interchanges.

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The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom