Partition interpretation of Gaussian binomial coefficients ($G_{n,k}(t)={\sum }_{\lambda \subseteq k\times (n-k)}{t}^{|\lambda |}$)

The Young diagrams of partitions in a rectangle with at most $k$ rows and at most $n-k$ cells in each row correspond to the lattice paths in the lattice-path interpretation of Gaussian binomial coefficients. Read the shaded row lengths from top to bottom; the result is a nonincreasing list of nonnegative integers. This bijection preserves the number $|\lambda |$ of cells.

In a two-by-two rectangle, the possibilities are the empty integer partition, $(1)$, $(2)$, $(1,1)$, $(2,1)$, and $(2,2)$. Their sizes are $0,1,2,2,3,4$, giving $G_{4,2}(t)$. In particular, there are two allowed diagrams of size two, explaining the middle coefficient.^[ref]

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 ${\lambda }_i^{\prime }=w-{\lambda }_{k+1-i}$. 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.