Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two New Interpretations of the Fishburn Numbers and their Refined
  Generating Functions   We show that two classes of combinatorial objects--inversion tables with no
subsequence of decreasing consecutive numbers and matchings with no
2-nestings--are enumerated by the Fishburn numbers. In particular, we give a
simple bijection between matchings with no 2-nestings and inversion tables with
no subsequence of decreasing consecutive numbers. We then prove using the
involution principle that inversion tables with no subsequence of decreasing
consecutive numbers have the same generating function as the Fishburn numbers.
The Fishburn numbers have previously been shown by Bousquet-M\'elou, Claesson,
Dukes and Kitaev to enumerate $\textbf{(2+2)}$-avoiding posets, matchings with
no left- or right-nestings, permutations avoiding a particular pattern, and
so-called ascent sequences, and by Dukes and Parviainen to enumerate upper
triangular matrices with non-negative entries and no empty rows or columns.
Claesson and Linusson conjectured they also enumerated matchings with no
2-nestings. Using these new interpretations of the Fishburn numbers and another
version of the involution, we prove the conjectured equality (also proven using
matrices by Jel\'{\i}nek and by Yan) of two refinements by Remmel and Kitaev of
the Fishburn generating function. In an appendix, we state and prove another
conjecture of Claesson and Linusson giving the distribution of left-nestings
over the set of all matchings.

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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ph
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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: eess
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci