## Abstract

A composition π=π_{1}π_{2}⋯π_{m} 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 π_{i}<πi+_{1}, π_{i}≤πi+_{1}, π_{i}=πi+_{1}, π_{i}>πi+_{1}, or π_{i}≥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 | American English |
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Pages (from-to) | 43-59 |

Number of pages | 17 |

Journal | Linear Algebra and Its Applications |

Volume | 449 |

DOIs | |

State | Published - 15 May 2014 |

## Keywords

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

## All Science Journal Classification (ASJC) codes

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