Identifier
-
Mp00202:
Integer partitions
—first row removal⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤ
Values
[1,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[2,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[1,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[3,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[2,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[2,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[4,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[3,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[3,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[2,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[5,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[4,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[4,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[3,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[6,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[5,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[5,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[4,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[7,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[6,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[6,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[5,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[8,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[7,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[7,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[6,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[9,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[8,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[8,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[7,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[10,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[9,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[9,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[8,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[11,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[10,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[10,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[9,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[12,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[11,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[11,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[10,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[13,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[12,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[12,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[11,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[14,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[13,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[13,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[12,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[15,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[14,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[14,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[13,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
[16,1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 0
[15,2] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1
[15,1,1] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 0
[14,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0
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searching the database for the individual values of this statistic
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
first row removal
Description
Removes the first entry of an integer partition
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.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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