Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The $\mathbf{uvu}$-avoiding $(a,b,c)$-Generalized Motzkin paths with
  vertical steps: bijections and statistic enumerations   A generalized Motzkin path, called G-Motzkin path for short, of length $n$ is
a lattice path from $(0, 0)$ to $(n, 0)$ in the first quadrant of the XOY-plane
that consists of up steps $\mathbf{u}=(1, 1)$, down steps $\mathbf{d}=(1, -1)$,
horizontal steps $\mathbf{h}=(1, 0)$ and vertical steps $\mathbf{v}=(0, -1)$.
An $(a,b,c)$-G-Motzkin path is a weighted G-Motzkin path such that the
$\mathbf{u}$-steps, $\mathbf{h}$-steps, $\mathbf{v}$-steps and
$\mathbf{d}$-steps are weighted respectively by $1, a, b$ and $c$. In this
paper, we first give bijections between the set of $\mathbf{uvu}$-avoiding
$(a,b,b^2)$-G-Motzkin paths of length $n$ and the set of $(a,b)$-Schr\"{o}der
paths as well as the set of $(a+b,b)$-Dyck paths of length $2n$, between the
set of $\{\mathbf{uvu, uu}\}$-avoiding $(a,b,b^2)$-G-Motzkin paths of length
$n$ and the set of $(a+b,ab)$-Motzkin paths of length $n$, between the set of
$\{\mathbf{uvu,uu}\}$-avoiding $(a,b,b^2)$-G-Motzkin paths of length $n+1$
beginning with an $\mathbf{h}$-step weighted by $a$ and the set of $(a,b)$-Dyck
paths of length $2n+2$. In the last section, we focus on the enumeration of
statistics "number of $\mathbf{z}$-steps" for $\mathbf{z}\in \{\mathbf{u},
\mathbf{h}, \mathbf{v}, \mathbf{d}\}$ and "number of points" at given level in
$\mathbf{uvu}$-avoiding G-Motzkin paths. These counting results are linked with
Riordan arrays.

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The categories of this paper are: physics
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-th
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: supr-con
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys