In 1857, while investigating the partition number function, J. J. Sylvester defined the function d(m; a1, . . . , an), called the denumerant, as the number of nonnegative integer representations of m by a1, . . . , an. This chapter is devoted to the study of the denumerant and related functions. After discussing briefly some basic properties of the partition function and its relation with denumerants, the general behaviour of d(m; a1, . . . , an) and its connection to g(a1, . . . , an) are analyzed. Two interesting methods for computing denumerants — one based on a decomposition of the rational fraction into partial fractions and another due to E. T. Bell — are described. An exact value of d(m; p, q) — first found by T. Popoviciu in 1953 — is proved, and the known results when n = 2 and n = 3 are summarized. The calculation of g(a1, . . . , an) by using Hilbert series via free resolutions, and the use of this approach to show an explicit formula for g(a1, a2, a3), are shown. The connection among denumerants, FP, and Ehrhart polynomial as well as two variants of d(m; a1, . . . , an) are discussed.
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