Arnold Knopfmacher ; Toufik Mansour
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Record statistics in integer compositions
dmtcs:2691 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2009,
DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)
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https://doi.org/10.46298/dmtcs.2691
Record statistics in integer compositionsConference paper
Authors: Arnold Knopfmacher 1; Toufik Mansour 2
0000-0003-1962-043X##0000-0001-8028-2391
Arnold Knopfmacher;Toufik Mansour
1 The John Knopfmacher Centre for Applicable Analysis and Number Theory [Johannesburg]
2 Department of Mathematics [Haïfa]
A compositionσ=a1a2…am of n is an ordered collection of positive integers whose sum is n. An element ai in σ is a strong (weak) record if ai>aj(ai≥aj) for all j=1,2,…,i−1. Furthermore, the position of this record is i. We derive generating functions for the total number of strong (weak) records in all compositions of n, as well as for the sum of the positions of the records in all compositions of n, where the parts ai belong to a fixed subset A of the natural numbers. In particular when A=N, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of n.