Standard Young Tableaux and Bijection to saturated chains in Young Lattice

Question

I'm reading Sagan's book The Symmetric Group and am quite confused.

I was under the assumption that any tableau with entries weakly increasing along a row and strictly increasing down a column would be considered standard Young tableau, e.g.

$$1\; 2$$ $$2\; 3$$ would be a standard Young tableau. But Sagan proposes that there is a simple bijection between standard Young tableaux and saturated $\emptyset-\lambda$ chains in the Young lattice. But this wouldn't make sense for the above tableau, since you could take both:

• $\emptyset \prec (1,0) \prec (1,1) \prec (2,1) \prec (2,2)$
• $\emptyset \prec (1,0) \prec (2,0) \prec (2,1) \prec (2,2)$

I believe I am missing something, can someone please clarify?

Answer 1

Actually your Young tableau corresponds to the chain $$\begin{array}{cccccc}\emptyset & \prec & \bullet & \prec & \bullet & \bullet & \prec & \bullet & \bullet \\ & & & & \bullet & & & \bullet & \bullet\end{array}$$ that is, $\emptyset \prec (1,0) \prec (2,1) \prec (2,2)$, which is not saturated.