q-Binomial Coefficient

The -binomial coefficient is a q-analog for the binomial coefficient, also called a Gaussian coefficient or a Gaussian polynomial. A -binomial coefficient is given by

(1)

where

(2)

is a q-series (Koepf 1998, p. 26). For ,

(3)

where is a q-factorial (Koepf 1998, p. 30). The -binomial coefficient can also be defined in terms of the q-brackets by

(4)

The -binomial is implemented in the Wolfram Language as QBinomial[n, m, q].

For , the -binomial coefficients turn into the usual binomial coefficient.

The special case

(5)

is sometimes known as the q-bracket.

The -binomial coefficient satisfies the recurrence equation

(6)

for all and , so every -binomial coefficient is a polynomial in . The first few -binomial coefficients are

			(7)
			(8)
			(9)
			(10)

From the definition, it follows that

(11)

Additional identities include

			(12)
			(13)

The -binomial coefficient can be constructed by building all -subsets of , summing the elements of each subset, and taking the sum

(14)

over all subset-sums (Kac and Cheung 2001, p. 19).

The -binomial coefficient can also be interpreted as a polynomial in whose coefficient counts the number of distinct partitions of elements which fit inside an rectangle. For example, the partitions of 1, 2, 3, and 4 are given in the following table.

	partitions
0
1
2
3
4

Of these, , , , , , and fit inside a box. The counts of these having 0, 1, 2, 3, and 4 elements are 1, 1, 2, 1, and 1, so the (4, 2)-binomial coefficient is given by

(15)

as above.