Statistics for 3-letter patterns with repetitions in compositions
Armend Shaban Shabani, Rexhep Gjergji
Abstract
A composition π=π1 π2
⋯πm of a positive integer n is
an ordered collection of one or more positive integers whose sum is
n. The number of summands, namely m, is
called the number of parts of π. Using linear
algebra, we determine formulas for generating functions that count
compositions of n with m parts, according to
the number of occurrences of the subword pattern τ,
and according to the sum, over all occurrences of τ,
of the first integers in their respective occurrences, where
τ is any pattern of length three with exactly 2
distinct letters.
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