# Standard Young Tableaux and Bijection to saturated chains in Young Lattice

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?

• I don't have Sagan's book handy, but the usual terminology is that a Young tableau is standard if the rows and columns are both strictly increasing and the filling is with the numbers $1, 2, \dots, |\lambda|$ each occurring exactly once. What you wrote down is usually called a semi-standard Young tableau. Unfortunately, different people use slightly different terminology. Anyway, the theorem is that standard Young tableau as defined in this comment biject to saturated chains from $\emptyset$ to $\lambda$ in Young's lattice. – Michael Joyce Apr 12 '13 at 19:46
• Yes this is what I was thinking. I just wasnt sure if my previous notion of standard Young tableaux was different from Sagan's or whether my understanding of the ordering of the Young Lattice was off. Youre response is encouraging though, thnk you – gone Apr 12 '13 at 19:52
• @Zak: You might want to check the source of the definition you were using. While there are many terms in use for what you describe (including simply "tableaux" or "genrelaized Young tableaux") I have never seen them called "standard" Young tableaux. Again "standard" can mean different things in different situations, but not this. See the Wikipedia entry. – Marc van Leeuwen Apr 13 '13 at 7:34
• @MarcvanLeeuwen Yes this was the confusion. I only needed latter parts of Sagan's book and so I just assumed that the definitions used were consistent among what I've seen elsewhere – gone Apr 14 '13 at 20:29

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.

• Ah yes, I saw this; but I was looking at bijections to saturated chains – gone Apr 14 '13 at 20:30
• I'll just mark this as correct anyway lol – gone Apr 14 '13 at 21:53