representationcount

A representationcount, often referred to as a representation count, is a function that assigns to a target object, typically an integer, the number of representations of that object by a fixed mathematical structure under specified criteria. The idea is to count how many distinct ways the object can be expressed within the given framework.

Common contexts include quadratic forms, sums of powers, partitions, and, in a broader sense, representations of an object in a combinatorial or algebraic system. In number theory, a typical setup fixes a quadratic form Q in k variables with integer coefficients and asks how many integer vectors x satisfy Q(x) = n. This count is denoted r_Q(n) or rep_Q(n). The value depends on n and on the chosen form Q.

Classic examples include the representations of integers as sums of two or four squares. For the sum of two squares, the representation count r_2(n) is given by r_2(n) = 4(d_1(n) − d_3(n)), where d_1(n) and d_3(n) count divisors of n congruent to 1 or 3 modulo 4, respectively. For four squares, Jacobi’s theorem gives r_4(n) = 8 ∑_{d|n, 4∤d} d. These formulas illustrate how representation counts can be expressed in terms of the arithmetic of n.

Generating functions play a key role: the sequence r_Q(n) often appears as coefficients in the theta series associated with Q. Representationcounts are useful in analytic number theory, coding theory, and combinatorics, providing a bridge between structural definitions and arithmetic properties.