Permutations avoiding 1324 and patterns in Łukasiewicz paths

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The class Av(1324), of permutations avoiding the pattern 1324, is one of the simplest sets of combinatorial objects to define that has, thus far, failed to reveal its enumerative secrets. By considering certain large subsets of the class, which consist of permutations with a particularly regular structure, we prove that the growth rate of the class exceeds 9.81. This improves on a previous lower bound of 9.47. Central to our proof is an examination of the asymptotic distributions of certain substructures in the Hasse graphs of the permutations. In this context, we consider occurrences of patterns in Łukasiewicz paths and prove that in the limit they exhibit a concentrated Gaussian distribution.
Original languageEnglish
Pages (from-to)105–122
Number of pages18
JournalJournal of the London Mathematical Society
Issue number1
Early online date16 Jun 2015
Publication statusPublished - 31 Aug 2015


  • permutations
  • permutation classes
  • Av(3124)
  • Łukasiewicz paths

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