Standard Young Tableaux

1. Definition

A standard (Young) tableau of a partition $\lambda \vdash n $ is a bijection between the positive integers $\{1,2,\dots, n\}$ and the cells (also known as boxes) of the Young diagram of $\lambda$, such that rows and columns are increasing. (English convention). A standard tableau of size $n$ is a standard tableau of a partition of size $n$.

2. Notation

By $\mathcal{ST}_n$, we denote the collection of all standard tableaux of $n$.

3. Examples

Here is another example from $\mathcal{ST}_9$.

$\begin{matrix}1 & 4 & 8\\2 & 5 & 9\\3 & 6\\7\\\end{matrix}$ which in Sage input form is $[[1,4,8], [2,5,9],[3,6],[7]]$.

4. Properties

5. Remarks

6. Statistics

We have the following 10 statistics in the database:

The charge of a standard tableau.
The number of attacking pairs of a standard tableau.
The number of inversions of a standard tableau.
The Shynar inversion number of a standard tableau.
The inversion number of a standard tableau as defined by Haglund and Stevens.
The orbit size of a standard tableau under promotion.
The number of descents of a standard tableau.
The cocharge of a standard tableau.
The major index of a standard tableau.
The leg major index of a standard tableau.

7. Maps

We have the following 6 maps in the database:

reading word permutation
to Gelfand-Tsetlin pattern
shape
conjugate
Schuetzenberger involution
catabolism

8. References

[Sch61]   C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13(1961), 179-191.

[Tym12]   Julianna Tymoczko, A simple bijection between standard 3×n tableaux and irreducible webs for sl3, J Algebr Comb (2012) 35:611–632.

9. Sage examples

StandardTableaux (last edited 2015-12-18 05:53:12 by ZachariahNeville)