Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting Dyck paths by area and rank   The set of Dyck paths of length $2n$ inherits a lattice structure from a
bijection with the set of noncrossing partitions with the usual partial order.
In this paper, we study the joint distribution of two statistics for Dyck
paths: \emph{area} (the area under the path) and \emph{rank} (the rank in the
lattice).
  While area for Dyck paths has been studied, pairing it with this rank
function seems new, and we get an interesting $(q,t)$-refinement of the Catalan
numbers. We present two decompositions of the corresponding generating
function: one refines an identity of Carlitz and Riordan; the other refines the
notion of $\gamma$-nonnegativity, and is based on a decomposition of the
lattice of noncrossing partitions due to Simion and Ullman.
  Further, Biane's correspondence and a result of Stump allow us to conclude
that the joint distribution of area and rank for Dyck paths equals the joint
distribution of length and reflection length for the permutations lying below
the $n$-cycle $(12...n)$ in the absolute order on the symmetric group.

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The categories of this paper are: physics
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The categories of this paper are: plasm-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: stat
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: q-alg