= Complement symmetry of Gaussian binomial coefficients
{title2=$G_{n,k}(t)=t^{k(n-k)}G_{n,k}(t^{-1})$}
Pad a <partition in a rectangle> with zero rows until it has $k$ entries, and put $w=n-k$. The complementary rotated diagram has row lengths $\lambda'_i=w-\lambda_{k+1-i}$. This is again a <partition in a rectangle>, and its size is $kw-|\lambda|$. Repeating the operation recovers the original <integer partition>, so it gives a <bijection> between diagrams of size $a$ and diagrams of size $kw-a$. Consequently the <coefficients> of the <Gaussian polynomial> read the same in either direction. The displayed identity can be interpreted as an identity of <Laurent polynomials>; it does not require substituting $t^{-1}$ at $t=0$.
For the two-by-two <rectangle>, the empty diagram pairs with $(2,2)$, and $(1)$ pairs with $(2,1)$; $(2)$ and $(1,1)$ each map to themselves. This establishes symmetry, while <unimodality of Gaussian binomial coefficients> is a stronger statement about the growth and decrease of successive <coefficients>.
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