高斯二項式係數

高斯二項式係數 (也稱作 高斯係數, 高斯多項式, 或 q-二項式係數)在數學里是指二項式係數的q-模擬。

定義

高斯二項式係數被定義為：

${\displaystyle {m \choose r}_{q}={\begin{cases}{\frac {(1-q^{m})(1-q^{m-1})\cdots (1-q^{m-r+1})}{(1-q)(1-q^{2})\cdots (1-q^{r})}}&r\leq m\\0&r>m\end{cases}}}$

其中， m 和 r 是非負整數。 當 r = 0時值為1。

高斯二項式係數計算一個有限維向量空間的子空間數。令q表示一個有限域里的元素數目，則在q元有限域上n維向量空間的k維子空間數等於

${\displaystyle {\binom {n}{k}}_{q}.}$

示例

${\displaystyle {0 \choose 0}_{q}={1 \choose 0}_{q}=1}$

${\displaystyle {1 \choose 1}_{q}={\frac {1-q}{1-q}}=1}$

${\displaystyle {2 \choose 1}_{q}={\frac {1-q^{2}}{1-q}}=1+q}$

${\displaystyle {3 \choose 1}_{q}={\frac {1-q^{3}}{1-q}}=1+q+q^{2}}$

${\displaystyle {3 \choose 2}_{q}={\frac {(1-q^{3})(1-q^{2})}{(1-q)(1-q^{2})}}=1+q+q^{2}}$

${\displaystyle {4 \choose 2}_{q}={\frac {(1-q^{4})(1-q^{3})}{(1-q)(1-q^{2})}}=(1+q^{2})(1+q+q^{2})=1+q+2q^{2}+q^{3}+q^{4}}$

性質

和普通二項式係數一樣, 高斯二項式係數是中心對稱的:

${\displaystyle {m \choose r}_{q}={m \choose m-r}_{q}.}$

特別地,

${\displaystyle {m \choose 0}_{q}={m \choose m}_{q}=1\,,}$

${\displaystyle {m \choose 1}_{q}={m \choose m-1}_{q}={\frac {1-q^{m}}{1-q}}=1+q+\cdots +q^{m-1}\quad m\geq 1\,.}$

當 q = 1 時，有

${\displaystyle {m \choose r}_{1}={m \choose r}}$

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