Gaussian binomial coefficient

A Gaussian binomial coefficient counts the subspaces of a given dimension in a finite-dimensional vector space over a finite field. It is a polynomial in a parameter that becomes an ordinary binomial coefficient when the parameter is set to 1.

Finite field: A field containing finitely many elements, with arithmetic performed modulo a prime or an irreducible polynomial. The coefficient counts subspaces over these fields, whose size supplies its parameter.
Vector space: A set of vectors with addition and scalar multiplication satisfying the axioms of linear algebra. The objects being counted are its subspaces.
Grassmannian: A space whose points represent linear subspaces of a fixed dimension in a vector space. Over a finite field, its points are counted by the Gaussian binomial coefficient.
q-analogue: A parameterized analogue of a mathematical object that recovers it in a suitable limit or specialization. Gaussian binomial coefficients are q-analogues of ordinary binomial coefficients.
Gaussian binomial theorem: An identity expanding products of terms involving powers of a variable using Gaussian binomial coefficients. It connects the coefficients’ product formula to their role in polynomial expansions.
Subspace: A subset of a vector space that is itself a vector space under the same operations. Each counted object is a subspace of the specified dimension.
Subspace lattice: The partially ordered set of subspaces of a vector space, ordered by inclusion. The coefficient gives the number of lattice elements at each dimension.
q-integer: The polynomial 1+q+⋯+qⁿ⁻¹, which specializes to the integer n at q=1. Products of q-integers provide a standard formula for Gaussian coefficients.
q-Pochhammer symbol: The finite product of factors of the form one minus a power of a parameter. This compact notation expresses the numerator and denominator of the coefficient’s product formula.
Dimension: The number of vectors in a basis of a vector space. The lower index specifies the dimension of the subspaces being counted.

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