Enumerating Segmented Patterns in Compositions and Encoding by Restricted Permutations

TL;DR

This paper develops generating functions to count segmented patterns in compositions, including palindromic ones, using encoding by restricted permutations, and explores bijections with other combinatorial objects.

Contribution

It introduces a new framework for enumerating segmented patterns in compositions via generating functions and restricted permutations, extending previous work on rises, drops, and levels.

Findings

Derived a generating function for segmented pattern occurrences in compositions.

Enumerated parts with specific sizes and positions in compositions and palindromic compositions.

Established bijections between restricted permutations and various combinatorial objects.

Abstract

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.