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)
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where
(2)
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is a q-series (Koepf 1998, p. 26). For ,
(3)
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where is a q-factorial (Koepf 1998, p. 30). The -binomial coefficient can also be defined in terms of the q-brackets by
(4)
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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)
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is sometimes known as the q-bracket.
The -binomial coefficient satisfies the recurrence equation
(6)
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for all and , so every -binomial coefficient is a polynomial in . The first few -binomial coefficients are
(7)
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(8)
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(9)
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(10)
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From the definition, it follows that
(11)
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Additional identities include
(12)
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(13)
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The -binomial coefficient can be constructed by building all -subsets of , summing the elements of each subset, and taking the sum
(14)
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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)
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as above.