Compositions of n satisfying some coprimality conditions

An ℓ-composition of n is a sequence of length ℓ of positive integers summing up to n. In this paper, we investigate the number of ℓ-compositions of n satisfying two natural coprimality conditions. Namely, we first give an exact asymptotic formula for the number of ℓ-compositions having the first summand coprime to the others. Then, we estimate the number of ℓ-compositions whose summands are all pairwise coprime.