= Young diagram
{c}
{title2=$Y(\lambda)$}
\Image[/young-diagram-hooks.png]
{title=The Young diagram of $(3,2)$, its conjugate $(2,2,1)$, a highlighted hook, and all five hook lengths}
The Young diagram of a <partition of an integer> $\lambda$ consists of one unit square, or cell, for each pair
$$
Y(\lambda)=\{(i,j):1\leq i\leq\ell(\lambda),\ 1\leq j\leq\lambda_i\}.
$$
Throughout this article we use the English convention: rows are left-aligned, row $1$ is at the top, row indices increase downwards, and column indices increase to the right. The row lengths weakly decrease from top to bottom. Thus $Y(3,2)$ has cells $(1,1),(1,2),(1,3),(2,1),(2,2)$, with three cells above two. In the French convention the longest row is drawn at the bottom; the underlying <partition of an integer> is unchanged, but directions in descriptions of <hooks of a Young diagram> and <standard Young tableaux> must be interpreted consistently. See https://www.cip.ifi.lmu.de/~grinberg/t/24s/hooktalk.pdf[Grinberg's hook-length-formula teaching slides, §§1.1–1.2, for diagrams with the same orientation].
The number of cells is $|\lambda|$, and reading the row lengths recovers the <partition of an integer>. Reading the column lengths instead gives its <conjugate partition>. This turns numerical inequalities into visible geometry: <Young diagrams> satisfy
$$
Y(\mu)\subseteq Y(\lambda)\quad\Longleftrightarrow\quad
\mu_i\leq\lambda_i\ \text{for every }i,
$$
with missing parts treated as zero. For example, $(2,1)\subseteq(3,2)$. This inclusion is the order in <Young's lattice>; it differs from <dominance order on partitions>, which compares sums of initial parts of <partitions of an integer> of the same size. The <partition of an integer> $(3,2)$ dominates $(2,2,1)$ because their successive sums are $3\geq2$, $5\geq4$, and $5=5$, but neither <Young diagram> contains the other: one has cell $(1,3)$, the other $(3,1)$.
The beauty of a <Young diagram> is that its shape simultaneously records a <partition of an integer>, indexes a <Specht module>, and supports local quantities such as <hook lengths> that determine a global <dimension>. Filling its cells leads to <Young tableaux>; adding one cell at a time leads to the <Young branching graph>.
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