Catalan continued fractions and increasing subsequences in permutations

Petter Brändén, Anders Claesson, Einar Steingrimsson

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Abstract

We call a Stieltjes continued fraction with monic monomial numerators a Catalan continued fraction. Let ek(π) be the number of increasing subsequences of length k+1 in the permutation π. We prove that any Catalan continued fraction is the multivariate generating function of a family of statistics on the 132-avoiding permutations, each consisting of a (possibly infinite) linear combination of the eks. Moreover, there is an invertible linear transformation that translates between linear combinations of eks and the corresponding continued fractions. Some applications are given, one of which relates fountains of coins to 132-avoiding permutations according to number of inversions. Another relates ballot numbers to such permutations according to number of right-to-left maxima.
Original languageEnglish
Pages (from-to)275–287
Number of pages13
JournalDiscrete Mathematics
Volume258
Issue number1-3
Early online date24 May 2002
DOIs
Publication statusPublished - 6 Dec 2002

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Keywords

  • Stieltjes continued fraction
  • Catalan continued fraction
  • permutations

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