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Compact, reversible mapping from set partitions of length k to integers

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Problem Detail: 

Given a set $S$ of length $n$, I'm looking to map all the $k$-length partitions of $S$ onto the set of integers such that these integers are as close to 0 as possible. Ideally the range would be $\left[0, {n \brace k}\right)$.

Ideally this mapping and its inverse would be easy to compute.

Asked By : Jordy Dickinson
Answered By : Yuval Filmus

The idea is to use the recurrence $$ \newcommand{\stirling}{\genfrac{\{}{\}}{0pt}{}} \stirling{n}{k} = k \stirling{n-1}{k} + \stirling{n-1}{k-1}. $$ This recurrence holds since either element $n$ belongs to one of the $k$ partitions of a partition of $1,\ldots,n-1$ into $k$ parts (say, ordered by minimal element), or it forms a singleton partition, joining a partition of $1,\ldots,n-1$ into $k-1$ parts.

Accordingly, partition the range $[0,\stirling{n}{k})$ into $k$ parts of length $\stirling{n-1}{k}$ and one of length $\stirling{n-1}{k-1}$. Given a partition, determine which of the $k+1$ intervals you belong to, remove $n$, encode the remaining partition of $n-1$ recursively, and locate it inside the chosen interval. The base cases are $\stirling{0}{0} = 1$ and $\stirling{n}{0} = \stirling{0}{n} = 0$ for $n > 0$.

The same algorithm can be reversed and be used to decode a number in $[0,\stirling{n}{k})$ into a partition of $1,\ldots,n$ into $k$ parts.

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