Abstract
A composition of the positive integer n is a representation of n as an ordered sum of positive integers $$n=a_1+a_2+\dots +a_m.$$ There are $$2^{n-1}$$ unrestricted compositions of n, which can be sorted according to the number of inversions they contain. (An inversion in a composition is a pair of summands $$\{a_i, a_j\}$$ for which $$ i< j$$ and $$a_i>a_j$$ .) The number of inversions of a composition is an indication of how far the composition is from a partition of n, which by convention uses a sequence of nondecreasing summands and thus has no inversions. We count compositions of n with exactly r inversions in several ways to derive generating function identities, and also consider asymptotic results.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
