Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Troupes, Cumulants, and Stack-Sorting   Several sequences of free cumulants that count binary plane trees correspond
to sequences of classical cumulants that count the decreasing versions of the
same trees. Using two new operations on colored binary plane trees that we call
insertion and decomposition, we prove that this surprising phenomenon holds for
families of trees that we call troupes. We give a simple characterization of
troupes, which provide a broad framework for generalizing several of the
results known about West's stack-sorting map $s$. Indeed, we give new proofs of
some of the main techniques that have been developed for understanding $s$;
these new proofs are far more conceptual than the original ones, explain how
the objects called valid hook configurations arise naturally, and generalize to
troupes. For $t\in\{2,3\}$, we enumerate $t$-stack-sortable alternating
permutations of odd length and $t$-stack-sortable permutations whose descents
are all peaks.
  The unexpected connection between troupes and cumulants provides a powerful
new tool for analyzing the stack-sorting map that hinges on free probability
theory. We give numerous applications of this method. For example, we show that
if $\sigma\in S_{n-1}$ is chosen uniformly at random, then the expected value
of $\text{des}(s(\sigma))+1$ is \[\left(3-\sum_{j=0}^n\frac{1}{j!}\right)n.\]
Furthermore, the variance of $\text{des}(s(\sigma))+1$ is asymptotically
$(2+2e-e^2)n$. We obtain similar results concerning the expected number of
descents of postorder readings of decreasing colored binary plane trees. We
also obtain improved estimates for $|s(S_n)|$ and an improved lower bound for
the degree of noninvertibility of $s$. We give two novel formulas that convert
from free to classical cumulants. The first is given by a sum over noncrossing
partitions, and the second is given by a sum over $231$-avoiding valid hook
configurations. We pose several open problems.

18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
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The categories of this paper are: solv-int