6 0. GAUSSIAN BINOMIAL COEFFICIENT S
for all n.
for integers n 0 by
! = 1 and [n]! =  • • • [n] for n 0.
One has then
for all a n 0.
[n\ • [a — n\ •
0.2. An easy calculation shows for all a and n with n 0
"a + l~
n - 1
This implies (by induction on n) that all Gaussian binomial coefficients are con-
tained in Z[v,
(For fixed n use induction on a for all a n; use 0.1(3) to deal
with a 0.)
has a unique involutory automorphism that maps v to
to v. Denote it by x »— • x. We have clearly [a] = [a] for all a G Z, hence
for all a and n. If we apply it to (1), then we get for all a and
a + 1
n - 1
An elementary calculation shows for all r 1 that
^(-i)V^- 1 )
2 = 0
If we apply the automorphism above, then we get also
^ ( _ ! ) ^ - ^ - i )
0.3. If A : is a ring (with identity) and q G k a unit in k, then there is a unique
— k with i; i- q (and 1 i- 1). All [a],
are contained in Z[v, v
]. We denote their images by [a]v=zq etc.; by abuse of
notation we shall often write just [a] = [a]v=q etc., if it is clear which q we consider.
Note that [a]v=i = a, [n]\JZ=1 = n!, and
Warning: The definition of the Gaussian binomial coefficients in 0.1 is not
the traditional one that you can find (e.g.) in [Macdonald], I, §2, ex. 3.
You get that old version, if you change the definition of [a] in 0.1(1) to
— l)/(v — l). In general, those "classical" Gaussian binomial coefficients
(also called g-binomial coefficients) differ from those here by a q power
(positive or negative).