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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000332
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000332: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 0
[[1,0],[0,1]]
=> 0
[[0,1],[1,0]]
=> 1
[[1,0,0],[0,1,0],[0,0,1]]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> 1
[[1,0,0],[0,0,1],[0,1,0]]
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> 2
[[0,1,0],[0,0,1],[1,0,0]]
=> 2
[[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 5
Description
The positive inversions of an alternating sign matrix.
This is defined as
$$\sum_{i > k,j < l} A_{ij}A_{kl} - \text{the number of negative ones in the matrix}.$$
After counter-clockwise rotation, this is also the number of osculations in the corresponding fan of Dyck paths.
Matching statistic: St000441
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00004: Alternating sign matrices —rotate clockwise⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 27%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 27%
Values
[[1]]
=> [[1]]
=> [[1]]
=> [1] => 0
[[1,0],[0,1]]
=> [[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1,3] => 0
[[0,1],[1,0]]
=> [[1,0],[0,1]]
=> [[1,1],[2]]
=> [3,1,2] => 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [4,2,5,1,3,6] => 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [5,2,6,1,3,4] => 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [4,3,5,1,2,6] => 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [5,3,6,1,2,4] => 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => 2
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => 2
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,2,5,9,1,3,6,10] => 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,2,5,10,1,3,6,7] => 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,2,6,9,1,3,4,10] => 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,2,6,10,1,3,4,7] => 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,2,6,7,1,3,4,8] => 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,2,7,10,1,3,4,5] => 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,2,7,8,1,3,4,5] => 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,3,5,9,1,2,6,10] => 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,3,5,10,1,2,6,7] => 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,3,6,9,1,2,4,10] => 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,3,6,10,1,2,4,7] => 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,3,7,10,1,2,4,5] => 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => 5
[[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]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,7,12,4,8,13,2,5,9,14,1,3,6,10,15] => ? = 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]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,7,13,4,8,14,2,5,9,15,1,3,6,10,11] => ? = 1
[[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]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,8,12,4,9,13,2,5,10,14,1,3,6,7,15] => ? = 1
[[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]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,8,13,4,9,14,2,5,10,15,1,3,6,7,11] => ? = 1
[[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]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,8,14,4,9,15,2,5,10,11,1,3,6,7,12] => ? = 2
[[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]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,9,13,4,10,14,2,5,11,15,1,3,6,7,8] => ? = 2
[[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]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> [[1,2,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,9,14,4,10,15,2,5,11,12,1,3,6,7,8] => ? = 3
[[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]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,7,12,5,8,13,2,6,9,14,1,3,4,10,15] => ? = 1
[[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]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,7,13,5,8,14,2,6,9,15,1,3,4,10,11] => ? = 2
[[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]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,8,12,5,9,13,2,6,10,14,1,3,4,7,15] => ? = 1
[[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]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,3,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,8,13,5,9,14,2,6,10,15,1,3,4,7,11] => ? = 1
[[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]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,3,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,8,14,5,9,15,2,6,10,11,1,3,4,7,12] => ? = 2
[[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]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,3,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,9,13,5,10,14,2,6,11,15,1,3,4,7,8] => ? = 2
[[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]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,-1,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,9,14,5,10,15,2,6,11,12,1,3,4,7,8] => ? = 3
[[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]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [11,9,12,5,10,13,2,6,7,14,1,3,4,8,15] => ? = 2
[[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]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,4],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [12,9,13,5,10,14,2,6,7,15,1,3,4,8,11] => ? = 2
[[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]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,4],[2,3,3,4],[3,4,5],[4,5],[5]]
=> [13,9,14,5,10,15,2,6,7,11,1,3,4,8,12] => ? = 2
[[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]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,3,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> [14,9,15,5,10,11,2,6,7,12,1,3,4,8,13] => ? = 3
[[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]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,3],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [12,10,13,5,11,14,2,6,7,15,1,3,4,8,9] => ? = 3
[[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]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,3,3],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ? => ? = 3
[[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]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,3,3],[2,3,3,4],[3,4,4],[4,5],[5]]
=> [14,10,15,5,11,12,2,6,7,13,1,3,4,8,9] => ? = 4
[[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]]
=> [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,8,12,6,9,13,2,7,10,14,1,3,4,5,15] => ? = 2
[[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]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,8,13,6,9,14,2,7,10,15,1,3,4,5,11] => ? = 2
[[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]]
=> [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,8,14,6,9,15,2,7,10,11,1,3,4,5,12] => ? = 3
[[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]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,9,13,6,10,14,2,7,11,15,1,3,4,5,8] => ? = 2
[[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]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> ? => ? = 3
[[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]]
=> [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [11,9,12,6,10,13,2,7,8,14,1,3,4,5,15] => ? = 3
[[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]]
=> [[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,4],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [12,9,13,6,10,14,2,7,8,15,1,3,4,5,11] => ? = 3
[[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]]
=> [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,4],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ? => ? = 3
[[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]]
=> [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,2,4],[2,3,3,4],[3,4,4],[4,5],[5]]
=> [14,9,15,6,10,11,2,7,8,12,1,3,4,5,13] => ? = 4
[[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]]
=> [[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ? => ? = 3
[[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]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,3,4],[3,4,5],[4,5],[5]]
=> [13,10,14,6,11,15,2,7,8,12,1,3,4,5,9] => ? = 3
[[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]]
=> [[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,3,4],[3,4,4],[4,5],[5]]
=> ? => ? = 4
[[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]]
=> [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,3,3],[3,4,5],[4,5],[5]]
=> [13,11,14,6,12,15,2,7,8,9,1,3,4,5,10] => ? = 4
[[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]]
=> [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,2,3],[2,3,3,3],[3,4,4],[4,5],[5]]
=> [14,11,15,6,12,13,2,7,8,9,1,3,4,5,10] => ? = 5
[[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]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,9,13,7,10,14,2,8,11,15,1,3,4,5,6] => ? = 3
[[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]]
=> [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,9,14,7,10,15,2,8,11,12,1,3,4,5,6] => ? = 4
[[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]]
=> [[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,3,5],[3,4,5],[4,5],[5]]
=> [12,10,13,7,11,14,2,8,9,15,1,3,4,5,6] => ? = 4
[[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]]
=> [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,3,4],[3,4,5],[4,5],[5]]
=> ? => ? = 4
[[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]]
=> [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,3,4],[3,4,4],[4,5],[5]]
=> [14,10,15,7,11,12,2,8,9,13,1,3,4,5,6] => ? = 5
[[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]]
=> [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,3,3],[3,4,5],[4,5],[5]]
=> [13,11,14,7,12,15,2,8,9,10,1,3,4,5,6] => ? = 5
[[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]]
=> [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> [[1,2,2,2,2],[2,3,3,3],[3,4,4],[4,5],[5]]
=> [14,11,15,7,12,13,2,8,9,10,1,3,4,5,6] => ? = 6
[[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]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,7,12,4,8,13,3,5,9,14,1,2,6,10,15] => ? = 1
[[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]]
=> [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,4,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,7,13,4,8,14,3,5,9,15,1,2,6,10,11] => ? = 2
[[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]]
=> [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,8,12,4,9,13,3,5,10,14,1,2,6,7,15] => ? = 2
[[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]]
=> [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,4],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,8,13,4,9,14,3,5,10,15,1,2,6,7,11] => ? = 2
[[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]]
=> [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,4],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,8,14,4,9,15,3,5,10,11,1,2,6,7,12] => ? = 3
[[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]]
=> [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [12,9,13,4,10,14,3,5,11,15,1,2,6,7,8] => ? = 3
[[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]]
=> [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[1,1,3,3,3],[2,3,4,4],[3,4,5],[4,5],[5]]
=> [13,9,14,4,10,15,3,5,11,12,1,2,6,7,8] => ? = 4
[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
=> [[1,1,2,4,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> [11,7,12,5,8,13,3,6,9,14,1,2,4,10,15] => ? = 1
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000454
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ? = 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ? = 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ([(0,1),(0,3),(1,2),(2,4),(3,5),(4,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,8),(1,5),(1,7),(2,4),(2,6),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,9),(7,9)],10)
=> ? = 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(0,10),(1,9),(1,12),(2,8),(2,11),(3,5),(3,6),(3,7),(4,5),(4,6),(4,13),(5,11),(6,12),(7,11),(7,12),(8,10),(8,13),(9,10),(9,13),(11,13),(12,13)],14)
=> ? = 3
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(0,2),(0,6),(1,2),(1,5),(3,4),(3,11),(4,9),(5,10),(6,8),(7,8),(7,14),(8,13),(9,11),(9,14),(10,12),(10,13),(11,12),(12,14),(13,14)],15)
=> ? = 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ([(0,3),(0,8),(1,10),(2,9),(3,11),(4,2),(5,4),(6,7),(7,1),(7,9),(8,5),(8,11),(9,10),(11,6)],12)
=> ([(0,5),(0,6),(1,4),(1,9),(2,3),(2,8),(3,10),(4,11),(5,8),(6,9),(7,10),(7,11),(8,10),(9,11)],12)
=> ? = 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ? = 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,6),(1,8),(2,9),(3,8),(3,9),(4,1),(5,4),(6,7),(7,2),(7,3),(8,10),(9,10)],11)
=> ([(0,1),(0,2),(1,4),(2,3),(3,8),(4,9),(5,7),(5,8),(6,7),(6,9),(7,10),(8,10),(9,10)],11)
=> ? = 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(0,8),(1,7),(2,5),(2,6),(3,7),(3,9),(4,8),(4,9),(5,7),(5,9),(6,8),(6,9)],10)
=> ? = 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(0,3),(0,4),(1,2),(1,9),(2,13),(3,10),(4,11),(5,10),(5,12),(6,9),(6,12),(7,11),(7,16),(8,14),(8,15),(9,13),(10,14),(11,15),(12,14),(13,16),(15,16)],17)
=> ? = 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> ([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> ([(0,1),(0,2),(1,4),(2,6),(3,5),(3,12),(4,8),(5,9),(6,10),(7,8),(7,12),(8,11),(9,10),(9,12),(10,11),(11,12)],13)
=> ? = 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ([(0,9),(0,11),(1,18),(2,17),(3,19),(4,13),(4,19),(5,12),(5,13),(6,16),(7,14),(8,5),(8,18),(9,10),(10,3),(10,4),(11,1),(11,8),(12,17),(13,15),(15,16),(16,14),(17,7),(18,2),(18,12),(19,6),(19,15)],20)
=> ([(0,15),(0,18),(1,14),(1,18),(2,16),(2,19),(3,17),(3,19),(4,6),(4,14),(5,7),(5,15),(6,16),(7,17),(8,9),(8,12),(8,13),(9,10),(9,11),(10,14),(10,18),(11,15),(11,18),(12,16),(12,19),(13,17),(13,19)],20)
=> ? = 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> ([(0,12),(0,13),(1,16),(2,15),(3,23),(4,19),(5,17),(5,20),(6,4),(7,5),(7,15),(7,16),(8,10),(9,7),(10,2),(11,1),(11,23),(12,8),(12,22),(13,14),(13,22),(14,3),(14,11),(15,17),(15,21),(16,20),(16,21),(17,24),(19,18),(20,19),(20,24),(21,24),(22,9),(23,6),(24,18)],25)
=> ([(0,1),(0,5),(1,22),(2,4),(2,12),(3,16),(3,20),(4,10),(5,13),(6,17),(6,23),(7,10),(7,21),(8,12),(8,18),(9,13),(9,19),(10,24),(11,20),(11,21),(11,22),(12,16),(13,17),(14,15),(14,21),(14,22),(15,23),(15,24),(16,18),(17,19),(18,19),(20,23),(20,24),(21,24),(22,23)],25)
=> ? = 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> ([(0,11),(1,4),(2,10),(2,14),(3,9),(3,13),(4,12),(5,6),(5,7),(5,15),(6,12),(6,13),(7,12),(7,14),(8,12),(8,13),(8,14),(9,11),(9,15),(10,11),(10,15),(13,15),(14,15)],16)
=> ? = 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(0,20),(0,28),(1,19),(1,27),(2,22),(2,29),(3,21),(3,23),(4,8),(4,11),(5,6),(5,12),(6,19),(7,8),(7,24),(9,10),(9,16),(9,17),(10,19),(10,27),(11,21),(11,22),(12,13),(12,20),(13,16),(13,30),(14,15),(14,29),(14,30),(15,24),(15,28),(16,26),(17,25),(17,27),(18,24),(18,26),(18,28),(20,30),(21,25),(22,25),(23,25),(23,27),(24,29),(26,29),(26,30),(28,30)],31)
=> ? = 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ([(0,5),(0,10),(1,16),(2,15),(3,14),(4,13),(5,12),(6,2),(6,13),(7,4),(7,14),(8,1),(9,6),(10,11),(10,12),(11,3),(11,7),(12,9),(13,15),(14,8),(15,16)],17)
=> ([(0,3),(0,4),(1,2),(1,9),(2,13),(3,10),(4,11),(5,10),(5,12),(6,9),(6,12),(7,11),(7,16),(8,14),(8,15),(9,13),(10,14),(11,15),(12,14),(13,16),(15,16)],17)
=> ? = 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> ([(0,2),(0,6),(1,2),(1,5),(3,4),(3,11),(4,9),(5,10),(6,8),(7,8),(7,14),(8,13),(9,11),(9,14),(10,12),(10,13),(11,12),(12,14),(13,14)],15)
=> ? = 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> ([(0,10),(0,12),(1,16),(2,17),(3,21),(4,22),(5,14),(6,13),(6,14),(7,13),(7,15),(8,7),(8,21),(9,2),(9,18),(10,11),(11,5),(11,6),(12,3),(12,8),(13,19),(14,9),(14,19),(15,22),(16,20),(17,20),(18,16),(18,17),(19,18),(21,4),(21,15),(22,1)],23)
=> ([(0,17),(0,22),(1,17),(1,18),(2,16),(2,19),(3,18),(3,19),(4,15),(4,22),(5,14),(5,21),(6,7),(6,14),(7,15),(8,9),(8,10),(8,11),(9,12),(9,13),(10,17),(10,22),(11,15),(11,22),(12,14),(12,21),(13,20),(13,21),(16,20),(16,21),(18,20),(19,20)],23)
=> ? = 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ([(0,14),(0,15),(1,19),(2,18),(3,29),(4,30),(5,22),(6,23),(7,24),(7,25),(8,9),(9,7),(9,18),(9,19),(10,5),(11,6),(12,2),(12,29),(13,1),(13,30),(14,16),(14,28),(15,17),(15,28),(16,3),(16,12),(17,4),(17,13),(18,24),(18,27),(19,25),(19,27),(20,26),(21,26),(22,20),(23,21),(24,22),(24,31),(25,23),(25,31),(27,31),(28,8),(29,10),(30,11),(31,20),(31,21)],32)
=> ([(0,15),(0,16),(1,18),(1,23),(2,17),(2,22),(3,24),(3,25),(4,20),(4,28),(5,21),(5,29),(6,19),(6,30),(7,9),(7,22),(8,10),(8,23),(9,11),(10,12),(11,24),(11,26),(12,25),(12,27),(13,26),(13,27),(13,30),(14,28),(14,29),(14,31),(15,17),(15,19),(16,18),(16,19),(17,20),(18,21),(20,22),(21,23),(24,31),(25,31),(26,28),(26,31),(27,29),(27,31),(28,30),(29,30)],32)
=> ? = 5
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ([(0,13),(0,15),(1,17),(2,16),(2,17),(3,19),(4,16),(4,18),(5,22),(6,21),(7,20),(8,23),(8,29),(9,4),(9,28),(10,3),(10,28),(11,6),(12,7),(12,24),(13,14),(14,1),(14,2),(15,9),(15,10),(16,25),(17,12),(17,25),(18,26),(18,29),(19,23),(19,26),(20,27),(22,27),(23,30),(24,20),(24,22),(25,24),(26,30),(27,21),(28,8),(28,18),(28,19),(29,5),(29,30),(30,11)],31)
=> ([(0,20),(0,28),(1,19),(1,27),(2,22),(2,29),(3,21),(3,23),(4,8),(4,11),(5,6),(5,12),(6,19),(7,8),(7,24),(9,10),(9,16),(9,17),(10,19),(10,27),(11,21),(11,22),(12,13),(12,20),(13,16),(13,30),(14,15),(14,29),(14,30),(15,24),(15,28),(16,26),(17,25),(17,27),(18,24),(18,26),(18,28),(20,30),(21,25),(22,25),(23,25),(23,27),(24,29),(26,29),(26,30),(28,30)],31)
=> ? = 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ([(0,5),(0,16),(0,17),(1,23),(2,19),(3,11),(3,21),(4,10),(4,20),(5,12),(5,13),(6,50),(7,51),(8,24),(8,58),(9,25),(9,59),(10,14),(10,52),(11,15),(11,53),(12,26),(12,60),(13,27),(13,60),(14,54),(15,55),(16,4),(16,29),(17,3),(17,29),(18,48),(18,49),(19,32),(19,33),(20,46),(20,52),(21,47),(21,53),(22,34),(22,35),(23,18),(23,54),(23,55),(24,40),(24,42),(25,41),(25,43),(26,46),(26,56),(27,47),(27,57),(28,63),(29,1),(30,62),(31,61),(32,61),(33,61),(34,6),(34,62),(35,7),(35,62),(36,58),(37,59),(38,32),(39,33),(40,44),(41,45),(42,38),(43,39),(44,31),(45,31),(46,36),(47,37),(48,40),(48,63),(49,41),(49,63),(50,38),(51,39),(52,8),(52,36),(53,9),(53,37),(54,28),(54,48),(55,28),(55,49),(56,30),(56,34),(57,30),(57,35),(58,42),(58,50),(59,43),(59,51),(60,22),(60,56),(60,57),(62,2),(63,44),(63,45)],64)
=> ([(0,43),(0,57),(1,42),(1,56),(2,45),(2,59),(3,44),(3,58),(4,46),(4,60),(5,47),(5,61),(6,8),(6,10),(6,11),(7,9),(7,12),(7,13),(8,22),(8,23),(9,24),(9,25),(10,18),(10,42),(11,19),(11,42),(12,20),(12,43),(13,21),(13,43),(14,28),(14,34),(14,36),(15,29),(15,35),(15,37),(16,30),(16,32),(16,38),(17,31),(17,33),(17,39),(18,28),(18,44),(19,29),(19,45),(20,30),(20,46),(21,31),(21,47),(22,34),(22,63),(23,35),(23,63),(24,32),(24,62),(25,33),(25,62),(26,58),(26,59),(26,62),(27,60),(27,61),(27,63),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,48),(36,50),(37,49),(37,51),(38,48),(38,50),(39,49),(39,51),(40,52),(40,53),(40,56),(41,54),(41,55),(41,57),(44,48),(45,49),(46,50),(47,51),(52,58),(52,62),(53,59),(53,62),(54,60),(54,63),(55,61),(55,63),(56,58),(56,59),(57,60),(57,61)],64)
=> ? = 6
[[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]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([],1)
=> ([],1)
=> 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]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[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]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[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]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[[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]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[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]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[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]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(2,5),(3,4),(4,6),(5,7)],8)
=> ? = 3
[[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]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[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]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[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]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[[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]]
=> [[1,1,1,1,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[[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]]
=> [[1,1,1,1,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 2
[[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]]
=> [[1,1,1,2,2],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ?
=> ?
=> ? = 2
[[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]]
=> [[1,1,1,2,3],[2,2,2,3],[3,3,4],[4,4],[5]]
=> ([(0,3),(0,7),(1,8),(2,8),(3,9),(4,5),(5,1),(6,2),(7,4),(7,9),(9,6)],10)
=> ([(0,1),(0,3),(1,2),(2,4),(3,5),(4,8),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 3
[[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]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[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]]
=> [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[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]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([],1)
=> ([],1)
=> 0
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
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