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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 20 statistics in the database:

St000009
Standard tableaux ⟶ ℤ
The charge of a standard tableau.
St000016
Standard tableaux ⟶ ℤ
The number of attacking pairs of a standard tableau.
St000017
Standard tableaux ⟶ ℤ
The number of inversions of a standard tableau.
St000057
Standard tableaux ⟶ ℤ
The Shynar inversion number of a standard tableau.
St000059
Standard tableaux ⟶ ℤ
The inversion number of a standard tableau as defined by Haglund and Stevens.
St000075
Standard tableaux ⟶ ℤ
The orbit size of a standard tableau under promotion.
St000157
Standard tableaux ⟶ ℤ
The number of descents of a standard tableau.
St000169
Standard tableaux ⟶ ℤ
The cocharge of a standard tableau.
St000330
Standard tableaux ⟶ ℤ
The (standard) major index of a standard tableau.
St000336
Standard tableaux ⟶ ℤ
The leg major index of a standard tableau.
St000507
Standard tableaux ⟶ ℤ
The number of ascents of a standard tableau.
St000508
Standard tableaux ⟶ ℤ
Eigenvalues of the random-to-random operator acting on a simple module.
St000693
Standard tableaux ⟶ ℤ
The modular (standard) major index of a standard tableau.
St000733
Standard tableaux ⟶ ℤ
The row containing the largest entry of a standard tableau.
St000734
Standard tableaux ⟶ ℤ
The last entry in the first row of a standard tableau.
St000735
Standard tableaux ⟶ ℤ
The last entry on the main diagonal of a standard tableau.
St000738
Standard tableaux ⟶ ℤ
The first entry in the last row of a standard tableau.
St000743
Standard tableaux ⟶ ℤ
The number of entries in a standard Young tableau such that the next integer is a....
St000744
Standard tableaux ⟶ ℤ
The length of the path to the largest entry in a standard Young tableau.
St000745
Standard tableaux ⟶ ℤ
The index of the last row whose first entry is the row number in a standard Young....

7. Maps

We have the following 12 maps in the database:

Mp00033
Dyck paths ⟶ Standard tableaux
to two-row standard tableau
Mp00042
Integer partitions ⟶ Standard tableaux
initial tableau
Mp00045
Integer partitions ⟶ Standard tableaux
reading tableau
Mp00059
Permutations ⟶ Standard tableaux
Robinson-Schensted insertion tableau
Mp00070
Permutations ⟶ Standard tableaux
Robinson-Schensted recording tableau
Mp00081
Standard tableaux ⟶ Permutations
reading word permutation
Mp00082
Standard tableaux ⟶ Gelfand-Tsetlin patterns
to Gelfand-Tsetlin pattern
Mp00083
Standard tableaux ⟶ Integer partitions
shape
Mp00084
Standard tableaux ⟶ Standard tableaux
conjugate
Mp00085
Standard tableaux ⟶ Standard tableaux
Schuetzenberger involution
Mp00106
Standard tableaux ⟶ Standard tableaux
catabolism
Mp00134
Standard tableaux ⟶ Binary words
descent word

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