Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The monomial basis and the $Q$-basis of the Hopf algebra of parking
  functions   Consider the vector space $\mathbb{K}\mathcal{P}$ spanned by parking
functions. By representing parking functions as labeled digraphs, Hivert,
Novelli and Thibon constructed a cocommutative Hopf algebra PQSym$^{*}$ on
$\mathbb{K}\mathcal{P}$. The product and coproduct of PQSym$^{*}$ are analogous
to the product and coproduct of the Hopf algebra NCSym of symmetric functions
in noncommuting variables defined in terms of the power sum basis. In this
paper, we view a parking function as a word. We shall construct a Hopf algebra
PFSym on $\mathbb{K}\mathcal{P}$ with a formal basis $\{M_a\}$ analogous to the
monomial basis of NCSym. By introducing a partial order on parking functions,
we transform the basis $\{M_a\}$ to another basis $\{Q_a\}$ via the M\"{o}bius
inversion. We prove the freeness of PFSym by finding two free generating sets
in terms of the $M$-basis and the $Q$-basis, and we show that PFSym is
isomorphic to the Hopf algebra PQSym$^{*}$. It turns out that our construction,
when restricted to permutations and non-increasing parking functions, leads to
a new way to approach the Grossman-Larson Hopf algebras of ordered trees and
heap-ordered trees.

19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: comp-gas
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The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: cs
0
The categories of this paper are: hep-ph