binomn

Binomn is a term encountered in some mathematical discussions to denote a generalized binomial coefficient associated with a vector of nonnegative integers. It is intended to extend the familiar binomial coefficient to multiple index components, offering a compact way to describe multiway combinatorial counts and polynomial expansions.

Definition and notation: For integers n ≥ 0 and r ≥ 2, and nonnegative integers k1, k2, ..., kr with k1 + k2 + ... + kr = n, binomn(n; k1, k2, ..., kr) is defined as n! / (k1! k2! ... kr!). This reduces to the ordinary binomial coefficient when r = 2 and (k1, k2) = (k, n − k). The coefficients are symmetric in the ki and are zero if any ki is negative or if the sum does not equal n.

Relation to the multinomial theorem: The binomn coefficients appear in the multinomial theorem, which states that (x1 + x2 + ... + xr)^n = sum over all ki with sum n of binomn(n; k1, ..., kr) ∏ xi^{ki}. This generalizes the binomial theorem and provides a compact expansion for multivariable polynomials.

Properties and applications: Binomn coefficients are integers and count sequences or arrangements of n items in r labeled categories with ki items of category i. They underpin probability distributions with multiple outcomes, combinatorial counting, and expansions in algebra and discrete mathematics. In many sources, binomn is viewed as another name for the multinomial coefficient, while some texts simply use multinomial coefficients without the binomn label.

See also: Binomial coefficient, Multinomial coefficient, Multinomial theorem. Example: binomn(5; 2, 1, 2) = 5! / (2! 1! 2!) = 30.