= Partition interpretation of Gaussian binomial coefficients
{title2=$G_{n,k}(t)=\sum_{\lambda\subseteq k\times(n-k)}t^{|\lambda|}$}
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, \b[there are two allowed diagrams of size two], explaining the middle <coefficient>.<Enumerative Combinatorics, Volume 1 (second-edition draft) by Richard Stanley>[§1.7, Proposition 1.7.3, pp. 64–66]{ref}
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