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26 Combinatorial AnalysisProperties

§26.17 The Twelvefold Way

The twelvefold way gives the number of mappings f from set N of n objects to set K of k objects (putting balls from set N into boxes in set K). See Table 26.17.1. In this table (k)n is Pochhammer’s symbol, and S(n,k) and pk(n) are defined in §§26.8(i) and 26.9(i).

Table 26.17.1 is reproduced (in modified form) from Stanley (1997, p. 33). See also Example 3 in §26.18.

Table 26.17.1: The twelvefold way.
elements of N elements of K f unrestricted f one-to-one f onto
labeled labeled kn (k-n+1)n k!S(n,k)
unlabeled labeled (k+n-1n) (kn) (n-1n-k)
labeled unlabeled S(n,1)+S(n,2)++S(n,k) {1nk0n>k S(n,k)
unlabeled unlabeled pk(n) {1nk0n>k pk(n)-pk-1(n)