Greatest Descents after any Maxima in Compositions

  • Margaret Archibald
  • Aubrey Blecher
  • Charlotte Brennan
  • Arnold Knopfmacher
Keywords: Compositions, Descents, Generating functions, Asymptotics, Mellin transforms


In this paper, compositions of $n$ are studied. These are sequences of positive integers $(\sigma_i)_{i=1}^k$ whose sum is $n$. We define a maximum to be a part which is greater than or equal to all other parts. We investigate the size of the descents immediately following any maximum and we focus particularly on the largest and average of these, obtaining the generating functions in each case. Using Mellin transforms, we obtain asymptotic expressions for these quantities.
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