Mathematical Sciences: Faculty Presentations (1991-2022)

Parts and Subword Patterns in Compositions

Document Type

Presentation

Presentation Date

9-12-2015

Copyright

This work is archived and distributed under the repository's Standard Copyright and Reuse License (opens in new tab). End users may copy, store, and distribute this work without restriction. For all other uses, permission must be obtained from the copyright owners or their authorized agents.

Abstract or Description

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

Link to Full Text

Share

COinS