Patterns and Parts in Compositions: Enumeration and Bijection
A composition of an integer n is a tuple of positive integers that sum up to n. Our study began with the empirical observation that, in the set of all compositions of n, the total number of odd parts equals the total number of runs. We explore proofs of this fact through combinatorial as well as generating function approaches. From there we show more general results relating the number of parts in a given residue class modulo m to various subword patterns among all compositions of n.
The Integers Conference
Wang, Hua, Brian Hopkins, Mark Shattuck, Andrew V. Sills, Thotsaporn Thanatipanonda.
"Patterns and Parts in Compositions: Enumeration and Bijection."
Mathematical Sciences Faculty Presentations.