Identifier
Values
[1] => [1,0] => [1] => [1] => 1
[2] => [1,0,1,0] => [1,2] => [2,1] => 1
[1,1] => [1,1,0,0] => [2,1] => [1,2] => 1
[3] => [1,0,1,0,1,0] => [1,2,3] => [3,2,1] => 1
[2,1] => [1,0,1,1,0,0] => [1,3,2] => [2,3,1] => 1
[1,1,1] => [1,1,0,1,0,0] => [2,3,1] => [1,3,2] => 2
[4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [4,3,2,1] => 1
[3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [3,4,2,1] => 1
[2,2] => [1,1,1,0,0,0] => [3,1,2] => [2,1,3] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => [2,4,3,1] => 2
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,4,3,2] => 5
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [4,5,3,2,1] => 1
[3,2] => [1,0,1,1,1,0,0,0] => [1,4,2,3] => [3,2,4,1] => 1
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [3,5,4,2,1] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => [3,1,4,2] => [2,4,1,3] => 2
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [2,5,4,3,1] => 5
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [1,5,4,3,2] => 14
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => 1
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [4,3,5,2,1] => 1
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [4,6,5,3,2,1] => 2
[3,3] => [1,1,1,0,1,0,0,0] => [3,4,1,2] => [2,1,4,3] => 4
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [3,5,2,4,1] => 2
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [3,6,5,4,2,1] => 5
[2,2,2] => [1,1,1,1,0,0,0,0] => [4,1,2,3] => [3,2,1,4] => 1
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [2,5,4,1,3] => 5
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [2,6,5,4,3,1] => 14
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [1,6,5,4,3,2] => 42
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => 1
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [3,2,5,4,1] => 4
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [4,6,3,5,2,1] => 2
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [2,5,1,4,3] => 10
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [4,3,2,5,1] => 1
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [3,6,5,2,4,1] => 5
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => 14
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [3,5,2,1,4] => 2
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,1,4,5,6,2] => [2,6,5,4,1,3] => 14
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => [2,7,6,5,4,3,1] => 42
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => 132
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [4,3,6,5,2,1] => 4
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [2,1,5,4,3] => 21
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [3,6,2,5,4,1] => 10
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [5,4,3,6,2,1] => 1
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [4,2,1,5,3] => 4
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,1,5,6,2] => [2,6,5,1,4,3] => 28
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,6,4] => [4,6,3,2,5,1] => 2
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,2,5,6,7,3] => [3,7,6,5,2,4,1] => 14
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [3,2,1,5,4] => 8
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,1,2,5,6,3] => [3,6,5,2,1,4] => 5
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [3,1,4,5,6,7,2] => [2,7,6,5,4,1,3] => 42
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [3,2,6,5,4,1] => 21
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,1,6,2] => [2,6,1,5,4,3] => 61
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [5,3,2,6,4,1] => 4
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,5,2,6,7,3] => [3,7,6,2,5,4,1] => 28
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [4,3,2,1,5] => 1
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,1,2,6,4] => [4,6,2,1,5,3] => 8
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [3,4,1,5,6,7,2] => [2,7,6,5,1,4,3] => 84
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,5,6,2,3,4] => [4,3,2,6,5,1] => 8
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,1,2,6,3] => [3,6,2,1,5,4] => 20
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [4,1,2,5,6,7,3] => [3,7,6,5,2,1,4] => 14
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,1,2] => [2,1,6,5,4,3] => 135
[5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,5,6,2,7,3] => [3,7,2,6,5,4,1] => 61
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,1,2,5] => [5,2,1,6,4,3] => 21
[4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [3,4,5,1,6,7,2] => [2,7,6,1,5,4,3] => 186
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => 1
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,6,4] => [4,6,3,2,1,5] => 2
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,1,2,4] => [4,2,1,6,5,3] => 42
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => 82
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [4,5,1,2,6,7,3] => [3,7,6,2,1,5,4] => 56
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,4,5,6,7,2,3] => [3,2,7,6,5,4,1] => 135
[5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [3,4,5,6,1,7,2] => [2,7,1,6,5,4,3] => 435
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,1,2,4,5] => [5,4,2,1,6,3] => 4
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [5,1,6,2,3,4] => [4,3,2,6,1,5] => 8
[2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [4,5,6,1,2,7,3] => [3,7,2,1,6,5,4] => 242
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,7,1,2] => [2,1,7,6,5,4,3] => 1000
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [5,6,1,2,3,4] => [4,3,2,1,6,5] => 16
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => 1
[2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [4,5,6,7,1,2,3] => [3,2,1,7,6,5,4] => 1101
[] => [] => [] => [] => 1
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Description
The number of alternating sign matrices whose left key is the permutation.
The left key of an alternating sign matrix was defined by Lascoux
in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.
Map
reverse
Description
Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.