Counting ℓ-letter subwords in compositions
Toufik Mansour, Basel Sirhan
Abstract
Let ℕ be the set of all positive integers and let
A be any ordered subset of ℕ.
Recently, Heubach and
Mansour enumerated the number of compositions of n with
m
parts in A that contain the subword τ
exactly r
times, where τ∈{111,112,221,123}. Our
aims
are (1) to generalize the above results, i.e., to enumerate the number
of
compositions of n with m parts in
A
that contain an ℓ-letter subword, and (2) to analyze
the
number of compositions of n with m parts
that
avoid an ℓ-letter pattern, for given
ℓ.
We use tools such as asymptotic analysis of generating functions
leading to Gaussian asymptotic.
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