Young diagram

Definition

A Young diagram is a finite set $D \subseteq \mathbb{N} \times \mathbb{N}$ with this property: if $(i, j) \in D$, $i' \le i$ and $j' \le j$, then $(i', j') \in D$. The elements of $D$ are the cells, or boxes, of the diagram. A cell $(i, j)$ is drawn as a unit square in row $i$ and column $j$, with rows numbered from the top.

Remarks

The rows of a Young diagram have weakly decreasing lengths $\lambda_1 \ge \lambda_2 \ge \dots$. Thus a Young diagram with $n$ cells is the same as a partition of $n$. This drawing convention is the English one. The French convention draws row $0$ at the bottom. Transposing a diagram, $(i, j) \mapsto (j, i)$, gives the diagram of the conjugate partition.

Explanations

L1 human

A Young diagram draws a partition as rows of boxes, left-aligned, with the longest row at the top. The partition $4 + 2 + 1$ gives rows of $4$, $2$ and $1$ boxes.