Identifier
Values
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => [1,1] => 0
[3,1,1] => [[3,3,3],[2,2]] => [2,2] => [2] => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1] => 0
[1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => [2] => 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => [1,1] => 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => [1,1] => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => [2,1] => 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => [2,2] => 1
[3,1,2] => [[4,3,3],[2,2]] => [2,2] => [2] => 1
[3,2,1] => [[4,4,3],[3,2]] => [3,2] => [2] => 1
[4,1,1] => [[4,4,4],[3,3]] => [3,3] => [3] => 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => [1,1] => 0
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => [2] => 1
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 0
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => [1,1] => 0
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => [1,1] => 0
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => [2,1] => 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => [2,2] => 1
[1,3,1,2] => [[4,3,3,1],[2,2]] => [2,2] => [2] => 1
[1,3,2,1] => [[4,4,3,1],[3,2]] => [3,2] => [2] => 1
[1,4,1,1] => [[4,4,4,1],[3,3]] => [3,3] => [3] => 0
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => [1,1,1,1] => 0
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 0
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => [1,1,1] => 0
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => [1,1,1] => [1,1] => 0
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => [2,1,1] => 1
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => [2,1,1] => [1,1] => 0
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => [3,1,1] => [1,1] => 0
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => [2,2,1] => 2
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => [2,2,1] => [2,1] => 1
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => [3,2,1] => [2,1] => 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => [3,3,1] => [3,1] => 1
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => [2,2,2] => 2
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => [2,2,2] => [2,2] => 1
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => [3,2,2] => [2,2] => 1
[3,1,3] => [[5,3,3],[2,2]] => [2,2] => [2] => 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => [3,3,2] => [3,2] => 2
[3,2,2] => [[5,4,3],[3,2]] => [3,2] => [2] => 1
[3,3,1] => [[5,5,3],[4,2]] => [4,2] => [2] => 1
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => [3,3,3] => [3,3] => 2
[4,1,2] => [[5,4,4],[3,3]] => [3,3] => [3] => 0
[4,2,1] => [[5,5,4],[4,3]] => [4,3] => [3] => 0
[5,1,1] => [[5,5,5],[4,4]] => [4,4] => [4] => 1
[1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1],[1,1,1]] => [1,1,1] => [1,1] => 0
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 0
[1,1,2,1,1,2] => [[3,2,2,2,1,1],[1,1,1]] => [1,1,1] => [1,1] => 0
[1,1,2,1,2,1] => [[3,3,2,2,1,1],[2,1,1]] => [2,1,1] => [1,1] => 0
[1,1,2,2,1,1] => [[3,3,3,2,1,1],[2,2,1]] => [2,2,1] => [2,1] => 1
[1,1,3,1,2] => [[4,3,3,1,1],[2,2]] => [2,2] => [2] => 1
[1,1,3,2,1] => [[4,4,3,1,1],[3,2]] => [3,2] => [2] => 1
[1,2,1,1,1,1,1] => [[2,2,2,2,2,2,1],[1,1,1,1,1]] => [1,1,1,1,1] => [1,1,1,1] => 0
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => [1,1,1] => 0
[1,2,1,1,2,1] => [[3,3,2,2,2,1],[2,1,1,1]] => [2,1,1,1] => [1,1,1] => 0
[1,2,1,2,1,1] => [[3,3,3,2,2,1],[2,2,1,1]] => [2,2,1,1] => [2,1,1] => 1
[1,2,1,2,2] => [[4,3,2,2,1],[2,1,1]] => [2,1,1] => [1,1] => 0
[1,2,1,3,1] => [[4,4,2,2,1],[3,1,1]] => [3,1,1] => [1,1] => 0
[1,2,2,1,1,1] => [[3,3,3,3,2,1],[2,2,2,1]] => [2,2,2,1] => [2,2,1] => 2
[1,2,2,1,2] => [[4,3,3,2,1],[2,2,1]] => [2,2,1] => [2,1] => 1
[1,2,2,2,1] => [[4,4,3,2,1],[3,2,1]] => [3,2,1] => [2,1] => 1
[1,2,3,1,1] => [[4,4,4,2,1],[3,3,1]] => [3,3,1] => [3,1] => 1
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]] => [2,2,2] => [2,2] => 1
[1,3,1,2,1] => [[4,4,3,3,1],[3,2,2]] => [3,2,2] => [2,2] => 1
[1,3,1,3] => [[5,3,3,1],[2,2]] => [2,2] => [2] => 1
[1,3,2,1,1] => [[4,4,4,3,1],[3,3,2]] => [3,3,2] => [3,2] => 2
[1,3,2,2] => [[5,4,3,1],[3,2]] => [3,2] => [2] => 1
[1,3,3,1] => [[5,5,3,1],[4,2]] => [4,2] => [2] => 1
[1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]] => [3,3,3] => [3,3] => 2
[1,4,1,2] => [[5,4,4,1],[3,3]] => [3,3] => [3] => 0
[1,4,2,1] => [[5,5,4,1],[4,3]] => [4,3] => [3] => 0
[2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]] => [1,1,1,1,1,1] => [1,1,1,1,1] => 0
[2,1,1,1,1,2] => [[3,2,2,2,2,2],[1,1,1,1,1]] => [1,1,1,1,1] => [1,1,1,1] => 0
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]] => [2,1,1,1,1] => [1,1,1,1] => 0
[2,1,1,2,1,1] => [[3,3,3,2,2,2],[2,2,1,1,1]] => [2,2,1,1,1] => [2,1,1,1] => 1
[2,1,1,2,2] => [[4,3,2,2,2],[2,1,1,1]] => [2,1,1,1] => [1,1,1] => 0
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]] => [3,1,1,1] => [1,1,1] => 0
[2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]] => [2,2,2,1,1] => [2,2,1,1] => 3
[2,1,2,1,2] => [[4,3,3,2,2],[2,2,1,1]] => [2,2,1,1] => [2,1,1] => 1
[2,1,2,2,1] => [[4,4,3,2,2],[3,2,1,1]] => [3,2,1,1] => [2,1,1] => 1
[2,1,2,3] => [[5,3,2,2],[2,1,1]] => [2,1,1] => [1,1] => 0
[2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]] => [3,3,1,1] => [3,1,1] => 2
[2,1,3,2] => [[5,4,2,2],[3,1,1]] => [3,1,1] => [1,1] => 0
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]] => [2,2,2,2,1] => [2,2,2,1] => 5
[2,2,1,1,2] => [[4,3,3,3,2],[2,2,2,1]] => [2,2,2,1] => [2,2,1] => 2
[2,2,1,2,1] => [[4,4,3,3,2],[3,2,2,1]] => [3,2,2,1] => [2,2,1] => 2
[2,2,1,3] => [[5,3,3,2],[2,2,1]] => [2,2,1] => [2,1] => 1
[2,2,2,1,1] => [[4,4,4,3,2],[3,3,2,1]] => [3,3,2,1] => [3,2,1] => 6
[2,2,2,2] => [[5,4,3,2],[3,2,1]] => [3,2,1] => [2,1] => 1
[2,2,3,1] => [[5,5,3,2],[4,2,1]] => [4,2,1] => [2,1] => 1
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]] => [3,3,3,1] => [3,3,1] => 8
[2,3,1,2] => [[5,4,4,2],[3,3,1]] => [3,3,1] => [3,1] => 1
[2,3,2,1] => [[5,5,4,2],[4,3,1]] => [4,3,1] => [3,1] => 1
[2,4,1,1] => [[5,5,5,2],[4,4,1]] => [4,4,1] => [4,1] => 2
[3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]] => [2,2,2,2,2] => [2,2,2,2] => 5
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]] => [3,2,2,2] => [2,2,2] => 2
[3,1,1,3] => [[5,3,3,3],[2,2,2]] => [2,2,2] => [2,2] => 1
[3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]] => [3,3,2,2] => [3,2,2] => 8
[3,1,2,2] => [[5,4,3,3],[3,2,2]] => [3,2,2] => [2,2] => 1
[3,1,3,1] => [[5,5,3,3],[4,2,2]] => [4,2,2] => [2,2] => 1
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]] => [3,3,3,2] => [3,3,2] => 16
[3,2,1,2] => [[5,4,4,3],[3,3,2]] => [3,3,2] => [3,2] => 2
>>> Load all 125 entries. <<<
[3,2,2,1] => [[5,5,4,3],[4,3,2]] => [4,3,2] => [3,2] => 2
[3,2,3] => [[6,4,3],[3,2]] => [3,2] => [2] => 1
[3,3,1,1] => [[5,5,5,3],[4,4,2]] => [4,4,2] => [4,2] => 4
[3,3,2] => [[6,5,3],[4,2]] => [4,2] => [2] => 1
[3,4,1] => [[6,6,3],[5,2]] => [5,2] => [2] => 1
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]] => [3,3,3,3] => [3,3,3] => 16
[4,1,1,2] => [[5,4,4,4],[3,3,3]] => [3,3,3] => [3,3] => 2
[4,1,2,1] => [[5,5,4,4],[4,3,3]] => [4,3,3] => [3,3] => 2
[4,1,3] => [[6,4,4],[3,3]] => [3,3] => [3] => 0
[4,2,1,1] => [[5,5,5,4],[4,4,3]] => [4,4,3] => [4,3] => 6
[4,2,2] => [[6,5,4],[4,3]] => [4,3] => [3] => 0
[4,3,1] => [[6,6,4],[5,3]] => [5,3] => [3] => 0
[5,1,1,1] => [[5,5,5,5],[4,4,4]] => [4,4,4] => [4,4] => 6
[6,1,1] => [[6,6,6],[5,5]] => [5,5] => [5] => 0
[2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]] => [2,2,2,2,2,2] => [2,2,2,2,2] => 14
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]] => [3,3,3,3,3] => [3,3,3,3] => 168
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]] => [4,4,4,4] => [4,4,4] => 182
[6,1,1,1] => [[6,6,6,6],[5,5,5]] => [5,5,5] => [5,5] => 18
[7,1,1] => [[7,7,7],[6,6]] => [6,6] => [6] => 1
[2,1,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]] => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1] => [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]] => [2,2,2,2,2,2,2] => [2,2,2,2,2,2] => 42
[7,1,1,1] => [[7,7,7,7],[6,6,6]] => [6,6,6] => [6,6] => 57
[8,1,1] => [[8,8,8],[7,7]] => [7,7] => [7] => 0
search for individual values
searching the database for the individual values of this statistic
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd..
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
inner shape
Description
The inner shape of a skew partition.
Map
first row removal
Description
Removes the first entry of an integer partition