A Young diagram is a graphical way of representing a partition, similar to its Ferrers graph. For comparison, the Ferrers graph and the Young diagram of the partition $5+4+2+1$ are shown below.

The 3D analogue of a Young diagram maybe be used to give a graphical representation of a plane partition.

One advantage of Young diagrams is that each lattice path through a grid of height $m$ and width $n$, say from lower left corner to top right corner with only up- or right steps allowed, corresponds to the Young diagram of a partition with $\leq m$ parts, and largest part $\leq n$.
Thus the total number of such lattice paths equals the total number of partitions into $\leq m$ parts, each part $\leq n$. The former number is easily computed, by a simple counting argument, to be $$\binom{m+n}{n} = \binom{m+n}{m}.$$ This is illustrated in the diagram below, where the lattice path in red through the $6 \times 4$ lattice shown marks out the boundary of the Young diagram of the partition $4+2+1$ (green).
It can be seen that any similar path with up- or right steps from the bottom left corner to top right corner corresponds to a unique partition into $\leq 4$ parts, with the largest part being $\leq 6$ (the path that goes bottom left $\to$ top left $\to$ top right and fences off no cells corresponds to the one (empty) partition of 0).

Young diagrams play a part in the definition of hook numbers and t-cores. They play an important role in other areas of combinatorics, and in other areas of mathematics outside of combinatorics.

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