Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Actions on permutations and unimodality of descent polynomials   We study a group action on permutations due to Foata and Strehl and use it to
prove that the descent generating polynomial of certain sets of permutations
has a nonnegative expansion in the basis $\{t^i(1+t)^{n-1-2i}\}_{i=0}^m$,
$m=\lfloor (n-1)/2 \rfloor$. This property implies symmetry and unimodality. We
prove that the action is invariant under stack-sorting which strengthens recent
unimodality results of B\'ona. We prove that the generalized permutation
patterns $(13-2)$ and $(2-31)$ are invariant under the action and use this to
prove unimodality properties for a $q$-analog of the Eulerian numbers recently
studied by Corteel, Postnikov, Steingr\'{\i}msson and Williams.
  We also extend the action to linear extensions of sign-graded posets to give
a new proof of the unimodality of the $(P,\omega)$-Eulerian polynomials of
sign-graded posets and a combinatorial interpretations (in terms of
Stembridge's peak polynomials) of the corresponding coefficients when expanded
in the above basis.
  Finally, we prove that the statistic defined as the number of vertices of
even height in the unordered decreasing tree of a permutation has the same
distribution as the number of descents on any set of permutations invariant
under the action. When restricted to the set of stack-sortable permutations we
recover a result of Kreweras.

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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int