Enumerating Segmented Patterns in Compositions and Encoding by Restricted Permutations

Details Enumerating Segmented Patterns in Compositions and Encoding by Restricted Permutations Enumerating Segmented Patterns in Compositions and Encoding by Restricted Permutations

2005-05-05

arxiv.org/abs/math/0505094v1

Mathematics

Combinatorics

12 pages, 1 figure

Scientific paper

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.

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