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Given positive integers n {\displaystyle n} and d {\displaystyle d} , the d {\displaystyle d} -th Macaulay representation of n {\displaystyle n} is an expression for n {\displaystyle n} as a sum of binomial coefficients:
Here, c 1 , … , c d {\displaystyle c_{1},\ldots ,c_{d}} is a uniquely determined, strictly increasing sequence of nonnegative integers known as the Macaulay coefficients. For any two positive integers n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} , n 1 {\displaystyle n_{1}} is less than n 2 {\displaystyle n_{2}} if and only if the sequence of Macaulay coefficients for n 1 {\displaystyle n_{1}} comes before the sequence of Macaulay coefficients for n 2 {\displaystyle n_{2}} in lexicographic order.
Macaulay coefficients are also known as the combinatorial number system.