On $n$-scrambled tuples and distributional chaos in a sequence

The main aim of the present paper is to study relations between $n$-scrambled tuples and their attraction-adherence properties with respect to various sequences of integers. This extends previous research on relations between chaos in the sense of Li and Yorke and distributional chaos with respect to a given sequence. Moreover, we construct a system which is $n$-distributionally chaotic but not $(n+1)$-chaotic.