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.

References

• Huneke, Craig; Swanson, Irena (2006), "Appendix 5", Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK: Cambridge University Press, ISBN 978-0-521-68860-4, MR 2266432
• Caviglia, Giulio (2005), "A theorem of Eakin and Sathaye and Green's hyperplane restriction theorem", Commutative Algebra: Geometric, Homological, Combinatorial and Computational Aspects, CRC Press, ISBN 978-1-420-02832-4
• Green, Mark (1989), "Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann", Algebraic Curves and Projective Geometry, Lecture Notes in Mathematics, vol. 1389, Springer, pp. 76–86, doi:10.1007/BFb0085925, ISBN 978-3-540-48188-1