- FindStat

Identifier

Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000616: Permutations ⟶ ℤ

Values

[[1]] => [1] => 0

[[1,0],[0,1]] => [1,2] => 0

[[0,1],[1,0]] => [2,1] => 2

[[1,0,0],[0,1,0],[0,0,1]] => [1,2,3] => 0

[[0,1,0],[1,0,0],[0,0,1]] => [2,1,3] => 2

[[1,0,0],[0,0,1],[0,1,0]] => [1,3,2] => 3

[[0,1,0],[1,-1,1],[0,1,0]] => [1,3,2] => 3

[[0,0,1],[1,0,0],[0,1,0]] => [3,1,2] => 6

[[0,1,0],[0,0,1],[1,0,0]] => [2,3,1] => 5

[[0,0,1],[0,1,0],[1,0,0]] => [3,2,1] => 8

[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => [1,2,3,4] => 0

[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => [2,1,3,4] => 2

[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => [1,3,2,4] => 3

[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => [1,3,2,4] => 3

[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => [3,1,2,4] => 6

[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]] => [2,3,1,4] => 5

[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => [3,2,1,4] => 8

[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => [1,2,4,3] => 4

[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => [2,1,4,3] => 6

[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => [1,2,4,3] => 4

[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => [1,2,4,3] => 4

[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]] => [2,1,4,3] => 6

[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]] => [2,1,4,3] => 6

[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]] => [2,1,4,3] => 6

[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => [1,4,2,3] => 8

[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]] => [1,4,2,3] => 8

[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]] => [1,4,2,3] => 8

[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => [4,1,2,3] => 12

[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]] => [2,4,1,3] => 10

[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]] => [2,4,1,3] => 10

[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]] => [4,2,1,3] => 14

[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => [1,3,4,2] => 7

[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]] => [1,3,4,2] => 7

[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]] => [3,1,4,2] => 10

[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]] => [1,3,4,2] => 7

[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]] => [3,1,4,2] => 10

[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => [1,4,3,2] => 11

[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => [1,4,3,2] => 11

[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]] => [1,4,3,2] => 11

[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => [4,1,3,2] => 15

[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]] => [1,4,3,2] => 11

[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]] => [1,4,3,2] => 11

[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]] => [4,1,3,2] => 15

[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]] => [3,4,1,2] => 14

[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]] => [4,3,1,2] => 18

[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]] => [2,3,4,1] => 9

[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]] => [3,2,4,1] => 12

[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]] => [2,4,3,1] => 13

[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]] => [2,4,3,1] => 13

[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]] => [4,2,3,1] => 17

[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]] => [3,4,2,1] => 16

[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => [4,3,2,1] => 20

[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [1,2,3,4,5] => 0

[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [2,1,3,4,5] => 2

[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [1,3,2,4,5] => 3

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [1,3,2,4,5] => 3

[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [3,1,2,4,5] => 6

[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [2,3,1,4,5] => 5

[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [3,2,1,4,5] => 8

[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,2,4,3,5] => 4

[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,1,4,3,5] => 6

[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,2,4,3,5] => 4

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,2,4,3,5] => 4

[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,1,4,3,5] => 6

[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,1,4,3,5] => 6

[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,1,4,3,5] => 6

[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,4,2,3,5] => 8

[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,4,2,3,5] => 8

[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [1,4,2,3,5] => 8

[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [4,1,2,3,5] => 12

[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,4,1,3,5] => 10

[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [2,4,1,3,5] => 10

[[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [4,2,1,3,5] => 14

[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,3,4,2,5] => 7

[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,3,4,2,5] => 7

[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [3,1,4,2,5] => 10

[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,3,4,2,5] => 7

[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [3,1,4,2,5] => 10

[[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,4,3,2,5] => 11

[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,4,3,2,5] => 11

[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,4,3,2,5] => 11

[[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [4,1,3,2,5] => 15

[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,4,3,2,5] => 11

[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [1,4,3,2,5] => 11

[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [4,1,3,2,5] => 15

[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [3,4,1,2,5] => 14

[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [4,3,1,2,5] => 18

[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => [2,3,4,1,5] => 9

[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => [3,2,4,1,5] => 12

[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [2,4,3,1,5] => 13

[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [2,4,3,1,5] => 13

[[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [4,2,3,1,5] => 17

[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [3,4,2,1,5] => 16

[[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [4,3,2,1,5] => 20

[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [1,2,3,5,4] => 5

[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [2,1,3,5,4] => 7

[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [1,3,2,5,4] => 8

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [1,3,2,5,4] => 8

[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [3,1,2,5,4] => 11

[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [2,3,1,5,4] => 10

[[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [3,2,1,5,4] => 13

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Description

The inversion index of a permutation.
The inversion index of a permutation $\sigma=\sigma_1\sigma_2\ldots\sigma_n$ is defined as
$$ \sum_{\mbox{inversion pairs } (\sigma_i,\sigma_j)} \sigma_i $$
where $(\sigma_i,\sigma_j)$ is an inversion pair if i < j and $\sigma_i > \sigma_j$.
This equals the sum of the entries in the corresponding descending plane partition; see [1].

Map

to left key permutation

Description

Return the permutation of the left key of an alternating sign matrix.
This was originally defined by Lascoux and then further studied by Aval [1].