Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Labeled binary trees, subarrangements of the Catalan arrangements, and
  Schur positivity   In 1995, the first author introduced a multivariate generating function {$G$}
that tracks the distribution of ascents and descents in labeled binary trees.
In addition to proving that $G$ is symmetric, he conjectured that $G$ is Schur
positive. We prove this conjecture by expanding $G$ positively in terms of
ribbon Schur functions. We obtain this expansion using a weight-preserving
bijection whose inverse is inspired by the Push-Glide algorithm of
Pr\'{e}ville-Ratelle and Viennot. In fact, this weight-preserving bijection
allows us to establish a stronger version of the first author's conjecture
showing that the generating function restricted to labeled binary trees with a
fixed canopy is still Schur positive.
  We also discuss applications in the setting of hyperplane arrangements. We
show that a certain specialization of $G$ equals the Frobenius characteristic
of the natural $\mathfrak{S}_n$-action on regions of the semiorder arrangement,
which we then expand in terms of {the Frobenius characteristics} of Foulkes
characters. We also construct an $\mathfrak{S}_n$-action on regions of the
Linial arrangement using a set of trees studied by Bernardi, and subsequently
compute the character of this action by employing Lagrange inversion. The
resulting expression generalizes Postnikov's formula for the number of regions
in the Linial arrangement. As a final application, we prove
$\gamma$-nonnegativity for the distribution of the number of right edges over
local binary search trees.

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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: acc-phys
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: eess