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?