#### Title

Parts and Subword Patterns in Compositions

#### Document Type

Presentation

#### Publication Date

9-12-2015

#### Abstract

A “composition” of an integer n is a tuple of positive integers that sum to n. Thus the set of all compositions of 4 is {(4), (31), (13), (22), (211), (121), (112), (1111)}. Each summand is called a “part” of the composition. Let OP (n) denote the number of odd parts among all compositions of n. Thus OP (4) = 14. By a “run” in a composition we mean a collection of adjacent equal parts. Thus the composition (22222111311) contains four runs. Let R(n) denote the number of runs among all compositions of n. Notice that R(4) = 14. Our study began with the empirical observations that OP (n) = R(n) and EP (n + 1) = OP (n) where EP (n) = the number of even parts among all compositions of n. From there we were able to prove more general results relating the number of parts in a given residue class modulo m to various subword patterns among all compositions of n.

#### Sponsorship/Conference/Institution

Palmetto Number Theory Series (PANTS)

#### Location

Atlanta, GA

#### Recommended Citation

Sills, Andrew V..
2015.
"Parts and Subword Patterns in Compositions."
*Mathematical Sciences Faculty Presentations*.
Presentation 457.

https://digitalcommons.georgiasouthern.edu/math-sci-facpres/457