Identifier
-
Mp00108:
Permutations
—cycle type⟶
Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤ
Values
[1] => [1] => [1,0,1,0] => [[1,3],[2,4]] => 2
[1,2] => [1,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 2
[2,1] => [2] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 2
[1,2,3] => [1,1,1] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 2
[1,3,2] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[2,1,3] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[2,3,1] => [3] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 2
[3,1,2] => [3] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 2
[3,2,1] => [2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 2
[1,2,4,3] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2
[1,3,2,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2
[1,3,4,2] => [3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2
[1,4,2,3] => [3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2
[1,4,3,2] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2
[2,1,3,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 2
[2,3,1,4] => [3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2
[2,4,3,1] => [3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2
[3,1,2,4] => [3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2
[3,2,1,4] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2
[3,2,4,1] => [3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 2
[4,1,3,2] => [3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2
[4,2,1,3] => [3,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2
[4,2,3,1] => [2,1,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2
[4,3,2,1] => [2,2] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 2
[1,2,4,5,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[1,2,5,3,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[1,3,2,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[1,3,4,2,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[1,3,5,4,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[1,4,2,3,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[1,4,3,5,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[1,4,5,2,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[1,5,2,4,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[1,5,3,2,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[1,5,4,3,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[2,1,3,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[2,1,4,3,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[2,1,4,5,3] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[2,1,5,3,4] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[2,1,5,4,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[2,3,1,4,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[2,3,1,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[2,4,3,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[2,4,5,1,3] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[2,5,3,4,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[2,5,4,3,1] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[3,1,2,4,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[3,1,2,5,4] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[3,2,1,5,4] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[3,2,4,1,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[3,2,5,4,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[3,4,1,2,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[3,4,1,5,2] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[3,4,5,2,1] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[3,5,1,2,4] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[3,5,1,4,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[3,5,4,1,2] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[4,1,3,2,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[4,1,5,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[4,2,1,3,5] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[4,2,3,5,1] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[4,2,5,1,3] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[4,3,2,1,5] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[4,3,2,5,1] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[4,3,5,1,2] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[4,5,1,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[4,5,2,1,3] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[4,5,3,1,2] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[5,1,3,4,2] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[5,1,4,3,2] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[5,2,1,4,3] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[5,2,3,1,4] => [3,1,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 2
[5,2,4,3,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[5,3,2,1,4] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[5,3,2,4,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[5,3,4,2,1] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[5,4,1,2,3] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[5,4,2,3,1] => [3,2] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 2
[5,4,3,2,1] => [2,2,1] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => 2
[1,3,2,5,6,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,3,2,6,4,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,3,4,2,6,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,3,5,6,2,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,3,6,5,4,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,4,2,3,6,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,4,5,2,6,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,4,5,6,3,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,4,6,2,3,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,4,6,5,2,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,5,2,6,3,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,5,4,3,6,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,5,4,6,2,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,5,6,2,4,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,5,6,3,2,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,6,2,5,4,3] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,6,4,3,2,5] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,6,4,5,3,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,6,5,2,3,4] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
[1,6,5,3,4,2] => [3,2,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => 2
>>> Load all 201 entries. <<<
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
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Map
cycle type
Description
The cycle type of a permutation as a 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.
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!