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1. Definition

A semistandard (Young) tableau of a partition $\lambda \vdash n $ is map from the cells (also known as boxes) of the Young diagram of $\lambda$, to the natural numbers, such that rows are weakly increasing and the columns are increasing. A semistandard tableau of size $n$ is a semistandard tableau of a partition of size $n$.

All standard (Young) tableaux are also semistandard. The set of semistandard tableaux are in bijection with Gelfand-Tsetlin patterns.

2. Examples

Here is an example from $\mathcal{SST}_7$.

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

3. Properties

4. Remarks

5. Statistics

We have the following 11 statistics in the database:

St000101
Semistandard tableaux ⟶ ℤ
The cocharge of a semistandard tableau.
St000102
Semistandard tableaux ⟶ ℤ
The charge of a semistandard tableau.
St000103
Semistandard tableaux ⟶ ℤ
The sum of the entries of a semistandard tableau.
St000112
Semistandard tableaux ⟶ ℤ
The depth of a semistandard tableau $T$ in the crystal $B(\lambda)$ where $\lambda$ is ....
St000116
Semistandard tableaux ⟶ ℤ
The major index of a semistandard tableau obtained by standardizing.
St000170
Semistandard tableaux ⟶ ℤ
The trace of a semistandard tableau.
St000173
Semistandard tableaux ⟶ ℤ
The segment statistic of a semistandard tableau.
St000174
Semistandard tableaux ⟶ ℤ
The flush statistic of a semistandard tableau.
St000736
Semistandard tableaux ⟶ ℤ
The last entry in the first row of a semistandard tableau.
St000737
Semistandard tableaux ⟶ ℤ
The last entry on the main diagonal of a semistandard tableau.
St000739
Semistandard tableaux ⟶ ℤ
The first entry in the last row of a semistandard tableau.

6. Maps

We have the following 7 maps in the database:

Mp00001
Alternating sign matrices ⟶ Semistandard tableaux
to semistandard tableau
Mp00036
Gelfand-Tsetlin patterns ⟶ Semistandard tableaux
to semistandard tableau
Mp00075
Semistandard tableaux ⟶ Permutations
reading word permutation
Mp00076
Semistandard tableaux ⟶ Gelfand-Tsetlin patterns
to Gelfand-Tsetlin pattern
Mp00077
Semistandard tableaux ⟶ Integer partitions
shape
Mp00078
Semistandard tableaux ⟶ Semistandard tableaux
Schuetzenberger involution
Mp00107
Semistandard tableaux ⟶ Semistandard tableaux
catabolism

7. References

8. Sage examples