Descents after maxima in compositions
Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher
Abstract
We consider compositions of n, i.e., sequences of
positive integers (or parts)
(σi)i=1k where
σ1+σ2+⋯+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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