= Standard Young tableau
{title2=$\operatorname{SYT}(\lambda)$}
= Standard tableau
{synonym}
\Image[/standard-young-tableaux.png]
{title=All five standard Young tableaux of shape $(3,2)$ in the English convention}
A standard Young tableau of shape $\lambda\vdash n$ is a <Young tableau> that uses $1,\ldots,n$ exactly once and strictly increases left to right along every row and top to bottom down every column. Formally, a <bijection> $T:Y(\lambda)\to\{1,\ldots,n\}$ is standard when
$$
T(i,j)<T(i,j+1),\qquad T(i,j)<T(i+1,j)
$$
whenever the indicated cells belong to the <Young diagram>. We write $\operatorname{SYT}(\lambda)$ for the <set> of such fillings and $f^\lambda=|\operatorname{SYT}(\lambda)|$ for its <cardinality>.
For $(3,2)$, the five <standard Young tableaux> are listed below, with each pair giving the top row followed by the bottom row:
$$
\begin{aligned}
T_1&=((1,2,3),(4,5)),&T_2&=((1,2,4),(3,5)),\\
T_3&=((1,2,5),(3,4)),&T_4&=((1,3,4),(2,5)),\\
T_5&=((1,3,5),(2,4)).
\end{aligned}
$$
There are no others: the upper-left entry must be $1$, and choosing the two bottom-row entries fixes both increasing row orders. Of the six choices of two entries from $\{2,3,4,5\}$, only bottom row $(2,3)$ fails, because its $3$ would lie below $4$. Hence $f^{(3,2)}=5$ by direct enumeration, independently of the <hook-length formula>.
The cells occupied by $1,\ldots,r$ form a <Young diagram>: every cell above or to the left of an occupied cell has a smaller label. Consequently a <standard Young tableau> is equivalently a path from the empty shape to $\lambda$ in the <Young branching graph>, adjoining one <addable node of a Young diagram> at each step. For $T_2$, the successive shapes are
$$
\varnothing,\ (1),\ (2),\ (2,1),\ (3,1),\ (3,2).
$$
Conversely, number the cell added at step $r$ by $r$ to recover the <standard Young tableau>. This path interpretation also makes a <standard Young tableau> a <linear extension of a partially ordered set> defined by the row and column requirements; see https://math.mit.edu/~rstan/algcomb/algcomb.pdf[Stanley's Algebraic Combinatorics, Chapter 8, on tableaux and paths in Young's lattice].
The usefulness goes beyond counting: the <standard polytabloid basis> of a <Specht module> $S^\lambda$ over the <complex numbers> is indexed by $\operatorname{SYT}(\lambda)$, so $\dim_{\mathbb C}S^\lambda=f^\lambda$. Each of the five pictured <standard Young tableaux> therefore labels one <basis> element of the <irreducible representation> $S^{(3,2)}$ of the <symmetric group> $S_5$, whose <dimension> over the <complex numbers> is five. The filled diagrams index the <basis>; the actual <vectors> are <polytabloids>. The <hook-length formula> computes this <dimension> without listing all the <standard Young tableaux>.
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