Binomialis

Binomialis is a term used in combinatorics and algebra to denote the Gaussian, or q-binomial, coefficients. For integers n ≥ 0 and 0 ≤ k ≤ n, and a parameter q (often treated as a formal variable), binomialis [n choose k]_q generalizes the ordinary binomial coefficients. As q approaches 1, [n choose k]_q tends to the usual binomial coefficient C(n,k).

Definition: [n choose k]_q = product_{i=0}^{k-1} (1 - q^{n-i}) / (1 - q^{i+1}). Equivalently, [n choose k]_q = (1 - q^n)(1 - q^{n-1})... (1 - q^{n-k+1}) / (1 - q^k)(1 - q^{k-1})... (1 - q). This family of polynomials in q specializes to C(n,k) at q = 1 and remains a polynomial with nonnegative integer coefficients for all n and k.

Properties and interpretations: [n choose k]_q is symmetric in k and n−k, i.e., [n choose k]_q = [n choose n−k]_q, and it is unimodal as a polynomial in q. It counts combinatorial objects in the q-analogue setting: the number of k-dimensional subspaces of an n-dimensional vector space over the finite field with q elements equals [n choose k]_q. It also appears in the q-binomial theorem, where for variables a and b satisfying ab = q ba, (a + b)^n = sum_{k=0}^n [n choose k]_q a^{n−k} b^k.

Context: in standard literature these objects are called Gaussian or q-binomial coefficients. “Binomialis” is an alternate naming used in some expository contexts or informal discussions. They occupy a central role in combinatorics, representation theory, and the theory of symmetric functions.