Set Partitions Match Restricted Growth Functions
Set partitions of can be matched to restricted growth functions . Each entry of such a function (or -vector) is at most one more than the maximum of the preceding entries.[more]
The following explains the matching.
Suppose the partition is . We write this more compactly as .
Suppose the blocks are in order of their least element. In this example those elements are and the blocks are in order.
To construct the restricted growth function, put below the indices given by block : . Put below the indices given by block : . Finally put below the indices given by block : . Then the restricted growth function is .
To go the other way, reverse the process. For example, if the growth function is , then block is because those are the indices for . Block is and block is . The blocks are ordered by least element: .[less]
D. Stanton and D. White, Constructive Combinatorics, New York: Springer–Verlag, 1986.