Knopfmacher, A.2025-04-232018-05Knopfmacher, A., Mays, M.E. & Wagner, S. Compositions with a fixed number of inversions. Aequat. Math. 93, 601–617 (2019). https://doi.org/10.1007/s00010-018-0563-60001-9054 (print)1420-8903 (online)10.1007/s00010-018-0563-6https://hdl.handle.net/10539/44841A composition of the positive integer n is a representation of n as an ordered sum of positive integers n = a1 + a2 + ··· + am. There are 2n−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 {ai, aj} for which i<j and ai > aj .) 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.en© Springer International Publishing AG, part of Springer Nature 2018.CompositionPartitionInversionArticleN/A