User:Emily McCullough/sandbox

In combinatorics, block walking is a method of representing combinations and sums of combinations graphically as sets of North-East lattice paths or block walks.

A lattice path is "a path composed of connected horizontal and vertical line segments, each passing between adjacent lattice points."

The horizontal and vertical line segments in a lattice path, called steps, correspond to the horizontal and vertical sides of city blocks. The lattice points correspond to the intersections of these blocks.

A lattice path where the steps are restricted to North (N) and East (E) steps is called a North-East (NE) Lattice Path. A block walk is another term for a (NE) lattice path. It is conventional to orient a block walk so that the starting point is located at the origin, $$(0,0)$$.

The number of block walks from the origin to the point $$(a,b)$$ in the lattice, where $$a,b \in \N$$, is equal to the binomial coefficient
 * $$ \binom {a+b}{a} = \binom{a+b}{b} $$.

Similarly, the number of block walks with exactly $$n$$ steps, $$k$$ of which are E (equivalently, $$n-k$$ of which are N) is equal to
 * $$ \binom {n}{k} = \binom{n}{n-k} $$.