Lattice-path interpretation of Gaussian binomial coefficients ($G_{n,k}(t)={\sum }_P{t}^{A(P)}$)

Use lattice paths with $k$ north steps and $n-k$ east steps, beginning at $(0,0)$. Let $A(P)$ be the sum of the horizontal coordinates of its north steps, equivalently the number of unit cells to their left inside the $k$-by-$(n-k)$ rectangle. A north step in zero-based word position $j_i$ has $i-1$ earlier north steps, so its horizontal coordinate is $j_i-(i-1)$. The Gaussian binomial coefficient recurrence therefore counts these lattice paths with weight $t^{A(P)}$.

For $n=4,k=2$, the words NNEE, NENE, NEEN, ENNE, ENEN, EENN have respective areas $0,1,2,2,3,4$. The word NEEN contributes $t^2$ because the two north steps occur at horizontal coordinates zero and two. These paths never move south or west; allowing such extra steps would change the collection being counted.