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
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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
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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
cycle type
Description
The cycle type of a permutation as a partition.
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