& {x_2 } &  \cdots  & {x_n }  \\
\end{array}} \right) = \left( {\begin{array}{*{20}c}
   {j}  \\
   {\left\{x_i\right\}_{i=1}^n}  \\
\end{array}} \right)
 \mbox{,}$$ where $(y_i)_{i=1}^n$ is $(x_i)_{i=1}^n$ put into a canonical order.[^6] The indices $i_1,\,i_2,\,\ldots,\,i_n$ are a permutation of the integers $1,\,2,\,\ldots,\,n$ and a deterministic function of $(x_i)_{i=1}^n$; when the $x_i$s are not distinct, we require any deterministic mechanism for choosing amongst the permutations such that (\[eq:transform\]) holds. The ordering is collapsed into a single variable $j$, which defines a chance variable $J$.

The decomposition into order and value can be interpreted as the generation of “transform coefficients.” Whether decomposing signals into low frequency and high frequency [@Dudley1939]; predictable and unpredictable [@Oliver1952]; style and content [@TenenbaumF2000]; object and texture [@YanS1977]; or dictionary and pattern [@OrlitskySZ2004], divide-and-conquer approaches have been used to good advantage in many source coding scenarios. Here, we are concerned with situations in which the order $J$ is irrelevant and hence allocated no bits. In contrast, allocating all the bits to $J$ yields permutation source codes [@BergerJW1972]. Other rate allocations are discussed in [@VarshneyG2006b].

If the joint distribution of $(X_i)_{i=1}^n$ is exchangeable, $J$ and $\left\{X_i\right\}_{i=1}^n$ are statistically independent chance variables [@JiTK1995]. Thus they could be coded separately without loss of 