Discrete Mathematics & Theoretical Computer Science, Vol 16, No 1 (2014)

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Descents after maxima in compositions

Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher


We consider compositions of n, i.e., sequences of positive integers (or parts) (σi)i=1k where σ12+⋯+σk=n. We define a maximum to be any part which is not less than any other part. The variable of interest is the size of the descent immediately following the first and the last maximum. Using generating functions and Mellin transforms, we obtain asymptotic expressions for the average size of these descents. Finally, we show with the use of a simple bijection between the compositions of n for n>1, that on average the descent after the last maximum is greater than the descent after the first.

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