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Reduced Stirling numbers of the second kind

Denote the n objects to partition by the integers 1, 2,..., n. Define the reduced Stirling numbers of the second kind, denoted S d (n, k) {\displaystyle S^{d}(n,k)}, to be the number of ways to partition the integers 1, 2,..., n into k nonempty subsets such that all elements in each subset have pairwise distance at least d. That is, for any integers i and j in a given subset, it is required that | i − j | ≥ d {\displaystyle |i-j|\geq d}. It has been shown that these numbers satisfy

(hence the name "reduced"). Observe (both by definition and by the reduction formula), that S 1 (n, k) = S (n, k) {\displaystyle S^{1}(n,k)=S(n,k)}, the familiar Stirling numbers of the second kind.