Identifier
-
Mp00128:
Set partitions
—to composition⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤ
Values
{{1}} => [1] => [1,0] => [[1],[2]] => 0
{{1,2}} => [2] => [1,1,0,0] => [[1,2],[3,4]] => 1
{{1},{2}} => [1,1] => [1,0,1,0] => [[1,3],[2,4]] => 0
{{1,2,3}} => [3] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 2
{{1,2},{3}} => [2,1] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
{{1,3},{2}} => [2,1] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
{{1},{2,3}} => [1,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
{{1},{2},{3}} => [1,1,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is St000508Eigenvalues of the random-to-random operator acting on a simple module..
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is St000508Eigenvalues of the random-to-random operator acting on a simple module..
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to composition
Description
The integer composition of block sizes of a set partition.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
For a set partition of $\{1,2,\dots,n\}$, this is the integer composition of $n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.
Map
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!