The.Algorithm.Design.Manual.Springer-Verlag.1998

Generating Partitions
Set partitions can be generated using techniques similar to integer partitions. Each set partition is
encoded as a restricted growth function, , where and , for
. Each distinct digit identifies a subset, or block, of the partition, while the growth condition
ensures that the blocks are sorted into a canonical order based on the smallest element in each block. For
example, the restricted growth function 0,1,1,2,0,3,1 defines the set partition {{1,5}, {2,3,7}, {4}, {6} }.
Since there is a one-to-one equivalence between set partitions and restricted growth functions, we can use
lexicographic order on the restricted growth functions to order the set partitions. Indeed, the fifteen
partitions of {1,2,3,4} listed above are sequenced according to the lexicographic order of their restricted
growth function (check it out).
To randomly generate set partitions, we use a similar counting strategy as with integer partitions. The
Stirling numbers of the second kind S(n,k) count the number of partitions of 1,...,n with exactly k blocks.
They are computed using the recurrence S(n,k) = S(n-1,k-1) + k*S(n-1,k) with the boundary conditions
S(n,n)=1. The reader is referred to the references and implementations for more details and code.
Implementations: The best source on generating combinatorial objects is Nijenhuis and Wilf [NW78],
who provide efficient Fortran implementations of algorithms to construct random and sequential integer
partitions, set partitions, compositions, and Young tableaux. See Section for details on ftp-ing these
programs.
An exciting WWW site developed by Frank Ruskey of the University of Victoria contains a wealth of
material on generating combinatorial objects of different types, including permutations, subsets,
partitions, and certain graphs. It is well worth checking this out at http://sue.csc.uvic.ca/~cos/.
Combinatorica [Ski90] provides Mathematica implementations of algorithms to construct random and
sequential integer partitions, compositions, strings, and Young tableaux, as well as to count and
manipulate these objects. See Section .
Algorithm 403 [BL77] of the Collected Algorithms of the ACM is a Fortran code for constructing integer
partitions with k parts. It is available from Netlib (see Section ). The Stanford GraphBase (see
Section ) also contains generators for constructing all integer partitions.
Notes: The standard references on combinatorial generation [NW78, RND77, Rus97] all present
algorithms for generating integer and/or set partitions. Andrews is the primary reference on integer
partitions and related topics.
Interestingly, the set of all 52 set partitions for n=5 appears in the form of Murasaki diagrams in the oldest
novel known, The Tale of Genji.
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