Macaulay representation of an integer

Given positive integers ${\displaystyle n}$ and ${\displaystyle d}$, the ${\displaystyle d}$-th Macaulay representation of ${\displaystyle n}$ is an expression for ${\displaystyle n}$ as a sum of binomial coefficients:

${\displaystyle n={\binom {c_{d}}{d}}+{\binom {c_{d-1}}{d-1}}+\cdots +{\binom {c_{2}}{2}}+{\binom {c_{1}}{1}}.}$

Here, ${\displaystyle c_{1},\ldots ,c_{d}}$ is a uniquely determined, strictly increasing sequence of nonnegative integers known as the Macaulay coefficients. For any two positive integers ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$, ${\displaystyle n_{1}}$ is less than ${\displaystyle n_{2}}$ if and only if the sequence of Macaulay coefficients for ${\displaystyle n_{1}}$ comes before the sequence of Macaulay coefficients for ${\displaystyle n_{2}}$ in lexicographic order.

Macaulay coefficients are also known as the combinatorial number system.