Macaulay representation decomposes any nonnegative integer p into a unique sum of k binomial coefficients, one component for each combination position: p = x₀C1 + x₁C2 + … + xₖ₋₁Ck. The representation provides a combinatorial analogue of factorial-base decomposition. section 6.3.3
Its practical role is ranking combinations in lexicographic order. A combination can be converted to its combination number, then mapped back to the corresponding position in an ordered list. section 6.3.3
Combination numbers
For combinations of four elements drawn from six, the ordered output begins 0:0123, 1:0124, 2:0134, and continues through 14:2345. The value is the combination's position in the lexicographically ordered list.
