Enumerating segmented patterns in compositions and encoding with restricted permutations

Sergey Kitaev, Tyrrell McAllister, T. Kyle Petersen

Research output: Contribution to journalArticlepeer-review


A composition of a nonnegative integer n is a sequence of positive integers whose sum is n. A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of occurrences of arbitrary segmented partially ordered patterns among compositions of n with a prescribed number of parts. These patterns generalize the notions of rises, drops, and levels studied in the literature. We also obtain results enumerating parts with given sizes and locations among compositions and palindromic compositions with a given number of parts. Our results are motivated by “encoding by restricted permutations,” a relatively undeveloped method that provides a language for describing many combinatorial objects. We conclude with some examples demonstrating bijections between restricted permutations and other objects.
Original languageEnglish
Article numberA34
Number of pages16
JournalIntegers: Electronic Journal of Combinatorial Number Theory
Publication statusPublished - 11 Feb 2006


  • segmented patterns
  • palindromic compositions
  • bijections


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