---
tag: 0K78D81EF7
kind: definition
name: Young diagram
msc: [05A17]
defines: 096AHMB3P4
uses: [0404R9P9J3]
sources:
  - url: https://en.wikipedia.org/wiki/Young_tableau
    locator: Diagrams
---

## 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|0404R9P9J3]] 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.