In mathematics, the Gaussian binomial (sometimes called the Gaussian coefficient, the q-binomial coefficient, or the Gaussian polynomial) is a q-analog of the binomial coefficients.

The Gaussian binomial coefficients are given by

The generalization theorem states that the Gaussian binomials really are generalizations of the ordinary binomial coefficients; by specialization we have

The Pascal identities are

and

The Newton binomial formulas are

and

Applications

Gaussian binomials occur in the counting of symmetric polynomial's and in the theory of partitions. They also play an important role in the enumerative theory of symmetric space's defined over a finite field. In particular for every finite field F_q with q elements the Gaussian binomial counts the number v_n,k;q of different k-dimensional subvector spaces of an n-dimensional vector space over F_q. For example the Gaussian binomial is the number of different lines in F_qⁿ.

References

• Eugene Mukhin, Symmetric Polynomials and Partitions (http://mathcircle.berkeley.edu/BMC3/SymPol.pdf) (undated, 2004 or earlier).
• Ratnadha Kolhatkar, Zeta function of Grassmann Varietes (http://www.math.mcgill.ca/goren/SeminarOnCohomology/GrassmannVarieties%20.pdf) (dated January 26, 2004)