# Enumeration of compositions according to the sum of the values of the first letters of the occurrences of a 2-letter pattern

Walaa Asakly, Toufik Mansour

Research output: Contribution to journalArticlepeer-review

## Abstract

A composition π=π1π2⋯πm of a positive integer n is an ordered collection of one or more positive integers whose sum is n. The number of summands, namely m, is called the number of parts of π. We say that π contains a rise, a weak-rise, a level, a descent, or a weak-descent at position i according to whether πi<πi+1, πi≤πi+1, πi=πi+1, πi>πi+1, or πi≥πi+1. Using linear algebra, we determine formulas for generating functions that count compositions of n with m parts, according to the numbers of rises, weak-rises, levels, descents, and weak-descents, and according to the sum, over all occurrences of the rises, weak-rises, levels, descents, and weak-descents, of the first integers in their respective occurrences.

Original language English 43-59 17 Linear Algebra and Its Applications 449 https://doi.org/10.1016/j.laa.2014.02.028 Published - 15 May 2014

## Keywords

• Cramer's method
• Descents
• Generating functions
• Levels
• Rises
• Weak-descents
• Weak-rises

## ASJC Scopus subject areas

• Algebra and Number Theory
• Numerical Analysis
• Geometry and Topology
• Discrete Mathematics and Combinatorics

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