1. The $n \times n$ lattice just means the product poset of two chains: $[0,n] \times [0,n]=\{(i,j) | 0 \leq i,j \leq n\}$ where $(i,j) \leq (i',j')$ if and only if $i \leq i'$ and $j \leq j'$.
2. A linear extension of a poset $P$ with $m$ elements is just a map $f: P \to \{1,...,m\}$ such that $x \leq y$ in $P$ implies $f(x) \leq f(y)$.
3. If we identify the elements of $[0,n] \times [0,n]$ with the boxes of a square Young diagram, then the condition that $f$ is a linear extension is exactly the same as the condition that the tableau formed by putting the number $f(x)$ in the box corresponding to $x$ is a Young tableau.

This is a special case of a much more general phenomenon. The essential fact is that Young's lattice $Y$ (the set of Young diagrams ordered by inclusion) is the lattice of order ideals of $\mathbb{N} \times \mathbb{N}$. Therefore linear extensions of some ideal $I$ in $\mathbb{N} \times \mathbb{N}$ (in this case $I=[0,n] \times [0,n]$) correspond to chains from $\emptyset$ to $I$ in $Y$. Such chains can naturally be identified with Young tableau by giving boxes the label of the first element of the chain which contains that box.

1. The $n \times n$ lattice just means the product poset of two chains: $[0,n] \times [0,n]=\{(i,j) | 0 \leq i,j \leq n\}$ where $(i,j) \leq (i',j')$ if and only if $i \leq i'$ and $j \leq j'$.
2. A linear extension of a poset $P$ with $m$ elements is just a bijection $f: P \to \{1,...,m\}$ such that $x \leq y$ in $P$ implies $f(x) \leq f(y)$.
3. If we identify the elements of $[0,n] \times [0,n]$ with the boxes of a square Young diagram, then the condition that $f$ is a linear extension is exactly the same as the condition that the tableau formed by putting the number $f(x)$ in the box corresponding to $x$ is a Young tableau.

This is a special case of a much more general phenomenon. The essential fact is that Young's lattice $Y$ (the set of Young diagrams ordered by inclusion) is the lattice of order ideals of $\mathbb{N} \times \mathbb{N}$. Therefore linear extensions of some ideal $I$ in $\mathbb{N} \times \mathbb{N}$ (in this case $I=[0,n] \times [0,n]$) correspond to maximal chains from $\emptyset$ to $I$ in $Y$. Such chains can naturally be identified with Young tableau by giving boxes the label of the first element of the chain which contains that box.

1. The $n \times n$ lattice just means the product poset of two chains: $[0,n] \times [0,n]=\{(i,j) | 0 \leq i,j \leq n\}$ where $(i,j) \leq (i',j')$ if and only if $i \leq i'$ and $j \leq j'$.
2. A linear extension of a poset $P$ with $m$ elements is just a map $f: P \to \{1,...,m\}$ such that $x \leq y$ in $P$ implies $f(x) \leq f(y)$.
3. If we identify the elements of $[0,n] \times [0,n]$ with the boxes of a square Young diagram, then the condition that $f$ is a linear extension is exactly the same as the condition that the tableau formed by putting the number $f(x)$ in the box corresponding to $x$ is a Young tableau.

1. The $n \times n$ lattice just means the product poset of two chains: $[0,n] \times [0,n]=\{(i,j) | 0 \leq i,j \leq n\}$ where $(i,j) \leq (i',j')$ if and only if $i \leq i'$ and $j \leq j'$.
2. A linear extension of a poset $P$ with $m$ elements is just a map $f: P \to \{1,...,m\}$ such that $x \leq y$ in $P$ implies $f(x) \leq f(y)$.
3. If we identify the elements of $[0,n] \times [0,n]$ with the boxes of a square Young diagram, then the condition that $f$ is a linear extension is exactly the same as the condition that the tableau formed by putting the number $f(x)$ in the box corresponding to $x$ is a Young tableau.

This is a special case of a much more general phenomenon. The essential fact is that Young's lattice $Y$ (the set of Young diagrams ordered by inclusion) is the lattice of order ideals of $\mathbb{N} \times \mathbb{N}$. Therefore linear extensions of some ideal $I$ in $\mathbb{N} \times \mathbb{N}$ (in this case $I=[0,n] \times [0,n]$) correspond to chains from $\emptyset$ to $I$ in $Y$. Such chains can naturally be identified with Young tableau by giving boxes the label of the first element of the chain which contains that box.