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Your data matches 90 different statistics following compositions of up to 3 maps.
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Matching statistic: St000332
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
∑i>k,j<lAijAkl−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 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> [1] => 0
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [3,1,2] => 1
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1,3] => 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => 2
[[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]]
=> [[1,1,3],[2,3],[3]]
=> [4,3,5,1,2,6] => 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [5,2,6,1,3,4] => 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [4,2,5,1,3,6] => 0
[[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]]
=> [10,8,9,5,6,7,1,2,3,4] => 6
[[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
[[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
[[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],[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],[0,0,1,0],[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
[[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
[[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]]
=> [9,8,10,5,6,7,1,2,3,4] => 5
[[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
[[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
[[0,1,0,0],[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
[[0,0,1,0],[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
[[0,1,0,0],[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
[[0,0,1,0],[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
[[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
[[0,1,0,0],[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],[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
[[0,0,0,1],[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],[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]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => 2
[[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
[[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],[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
[[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]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => 3
[[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
[[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
[[0,1,0,0],[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],[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
[[0,0,0,1],[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],[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,-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
[[0,0,0,1],[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
[[0,0,1,0],[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
[[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],[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
[[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],[0,0,0,1],[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,-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
[[0,0,0,1],[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
Description
The number of successions of a permutation.
A succession of a permutation π is an index i such that π(i)+1=π(i+1). Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000796
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000796: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000796: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => [1] => ? = 0
[[1,0],[0,1]]
=> [1,2] => [2,1] => [2,1] => 1
[[0,1],[1,0]]
=> [2,1] => [1,2] => [1,2] => 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [2,3,1] => [3,1,2] => 2
[[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => [3,2,1] => [3,2,1] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [2,1,3] => [2,1,3] => 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [2,1,3] => [2,1,3] => 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [3,1,2] => [1,3,2] => 2
[[0,1,0],[0,0,1],[1,0,0]]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,3,2] => [2,3,1] => 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [3,2,4,1] => [4,2,1,3] => 4
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [2,4,3,1] => [4,2,3,1] => 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [2,4,3,1] => [4,2,3,1] => 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => [3,4,2,1] => [4,3,1,2] => 5
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [2,3,1,4] => [4,2,3,1] => [4,1,3,2] => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 6
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [2,4,1,3] => [1,3,4,2] => 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [2,4,1,3] => [1,3,4,2] => 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [2,4,1,3] => [1,3,4,2] => 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [3,4,1,2] => [1,4,2,3] => 4
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => [4,2,1,3] => [3,1,4,2] => 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => [4,2,1,3] => [3,1,4,2] => 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [4,2,1,3] => [4,3,1,2] => [1,4,3,2] => 5
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [3,1,4,2] => [3,1,2,4] => [1,3,2,4] => 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [3,1,4,2] => [3,1,2,4] => [1,3,2,4] => 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [3,1,4,2] => [3,4,1,2] => 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [3,1,4,2] => [3,4,1,2] => 2
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => [4,1,2,3] => [1,2,4,3] => 2
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [4,1,3,2] => [2,4,3,1] => 4
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [3,2,4,1] => [1,3,2,4] => [2,3,1,4] => 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => [1,2,4,3] => [2,3,4,1] => 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => [1,2,4,3] => [2,3,4,1] => 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [4,2,3,1] => [1,3,4,2] => [2,4,1,3] => 3
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [3,4,2,1] => [1,4,2,3] => [2,1,4,3] => 3
Description
The stat' of a permutation.
According to [1], this is the sum of the number of occurrences of the vincular patterns (13_2), (31_2), (32_2) and (21_), where matches of the underlined letters must be adjacent.
Matching statistic: St000798
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000798: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000798: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => [1] => ? = 0
[[1,0],[0,1]]
=> [1,2] => [2,1] => [2,1] => 1
[[0,1],[1,0]]
=> [2,1] => [1,2] => [1,2] => 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [2,3,1] => [3,1,2] => 2
[[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => [3,2,1] => [3,2,1] => 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [2,1,3] => [2,1,3] => 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [2,1,3] => [2,1,3] => 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [3,1,2] => [1,3,2] => 2
[[0,1,0],[0,0,1],[1,0,0]]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [1,3,2] => [2,3,1] => 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [3,2,4,1] => [4,2,1,3] => 4
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [2,4,3,1] => [4,2,3,1] => 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [2,4,3,1] => [4,2,3,1] => 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => [3,4,2,1] => [4,3,1,2] => 5
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [2,3,1,4] => [4,2,3,1] => [4,1,3,2] => 5
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 6
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 3
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 3
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [2,4,1,3] => [1,3,4,2] => 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [2,4,1,3] => [1,3,4,2] => 2
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [2,4,1,3] => [1,3,4,2] => 2
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [3,4,1,2] => [1,4,2,3] => 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => [4,2,1,3] => [3,1,4,2] => 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => [4,2,1,3] => [3,1,4,2] => 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [4,2,1,3] => [4,3,1,2] => [1,4,3,2] => 5
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [3,1,4,2] => [3,1,2,4] => [1,3,2,4] => 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [3,1,4,2] => [3,1,2,4] => [1,3,2,4] => 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 3
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [3,1,4,2] => [3,4,1,2] => 2
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [2,1,4,3] => [3,2,4,1] => 3
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [3,1,4,2] => [3,4,1,2] => 2
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => [4,1,2,3] => [1,2,4,3] => 3
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [4,1,3,2] => [2,4,3,1] => 4
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [3,2,4,1] => [1,3,2,4] => [2,3,1,4] => 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => [1,2,4,3] => [2,3,4,1] => 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => [1,2,4,3] => [2,3,4,1] => 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [4,2,3,1] => [1,3,4,2] => [2,4,1,3] => 2
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [3,4,2,1] => [1,4,2,3] => [2,1,4,3] => 4
Description
The makl of a permutation.
According to [1], this is the sum of the number of occurrences of the vincular patterns (132_), (31_2), (32_1) and (21_), where matches of the underlined letters must be adjacent.
Matching statistic: St001232
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 63%●distinct values known / distinct values provided: 57%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 63%●distinct values known / distinct values provided: 57%
Values
[[1]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[[1,0],[0,1]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[[0,1],[1,0]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,3}
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,3}
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [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]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,2,2,4,4,4,4,4,4,4,5,5,5,6}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000939
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 62%●distinct values known / distinct values provided: 57%
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 62%●distinct values known / distinct values provided: 57%
Values
[[1]]
=> [1,0]
=> [1,0]
=> []
=> ? = 0
[[1,0],[0,1]]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1}
[[0,1],[1,0]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1}
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,2,3}
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,2,3}
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,2,3}
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2,4,4,4,4,5,5,5,6}
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000777
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 57% ●values known / values provided: 62%●distinct values known / distinct values provided: 57%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 57% ●values known / values provided: 62%●distinct values known / distinct values provided: 57%
Values
[[1]]
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
[[1,0],[0,1]]
=> [1,2] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[0,1],[1,0]]
=> [2,1] => [1,2] => ([],2)
=> ? = 0 + 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,1,3} + 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,1,3} + 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,1,3} + 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 3 + 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [2,3,1,4] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [2,4,1,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 4 = 3 + 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [3,1,4,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,3,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [3,1,4,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [3,4,1,2] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [4,3,1,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [3,2,4,1] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [2,4,3,1] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [4,2,3,1] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [3,4,2,1] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001500
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001500: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 60%●distinct values known / distinct values provided: 57%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001500: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 60%●distinct values known / distinct values provided: 57%
Values
[[1]]
=> [1,0]
=> []
=> []
=> ? = 0
[[1,0],[0,1]]
=> [1,0,1,0]
=> [1]
=> [1,0]
=> ? ∊ {0,1}
[[0,1],[1,0]]
=> [1,1,0,0]
=> []
=> []
=> ? ∊ {0,1}
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,3}
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,3}
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,1,1,3}
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {0,1,1,3}
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 2
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 2
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 2
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 2
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 2
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0]
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,3,3,3,3,3,3,3,4,4,5,5,5,6}
Description
The global dimension of magnitude 1 Nakayama algebras.
We use the code below to translate them to Dyck paths.
Matching statistic: St001581
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 57%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 57%
Values
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0],[0,1]]
=> [[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[0,1],[1,0]]
=> [[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 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)
=> 2 = 1 + 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)
=> 3 = 2 + 1
[[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)
=> 3 = 2 + 1
[[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
[[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)
=> 1 = 0 + 1
[[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)
=> 2 = 1 + 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)
=> 2 = 1 + 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)
=> 2 = 1 + 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)
=> 3 = 2 + 1
[[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)
=> 3 = 2 + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> 2 = 1 + 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 + 1
[[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)
=> 2 = 1 + 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)
=> 3 = 2 + 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)
=> 3 = 2 + 1
[[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)
=> 3 = 2 + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> 3 = 2 + 1
[[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)
=> 4 = 3 + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> 4 = 3 + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> 3 = 2 + 1
[[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)
=> 3 = 2 + 1
[[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)
=> 4 = 3 + 1
[[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)
=> 4 = 3 + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
[[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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 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]]
=> ([(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)
=> 4 = 3 + 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]]
=> ([(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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 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]]
=> ([(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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 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]]
=> ([(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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 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]]
=> ([(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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 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]]
=> ([(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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 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]]
=> ([(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)
=> ? ∊ {2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6} + 1
Description
The achromatic number of a graph.
This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St000260
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 57%
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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> 2
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> 3
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
[[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)
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,6}
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 80 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000172The Grundy number of a graph. St000918The 2-limited packing number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000454The largest eigenvalue of a graph if it is integral. St001812The biclique partition number of a graph. St000456The monochromatic index of a connected graph. St000259The diameter of a connected graph. St001060The distinguishing index of a graph. St000028The number of stack-sorts needed to sort a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St001118The acyclic chromatic index of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000422The energy of a graph, if it is integral. St000996The number of exclusive left-to-right maxima of a permutation. St000665The number of rafts of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000174The flush statistic of a semistandard tableau. St000072The number of circled entries. St000073The number of boxed entries. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000155The number of exceedances (also excedences) of a permutation. St000173The segment statistic of a semistandard tableau. St000214The number of adjacencies of a permutation. St000223The number of nestings in the permutation. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3. St000710The number of big deficiencies of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000836The number of descents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001403The number of vertical separators in a permutation. St001631The number of simple modules S with dimExt1(S,A)=1 in the incidence algebra A of the poset. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001052The length of the exterior of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001415The length of the longest palindromic prefix of a binary word. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000646The number of big ascents of a permutation. St000668The least common multiple of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St001498The normalised height of a Nakayama algebra with magnitude 1. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001330The hat guessing number of a graph. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001271The competition number of a graph. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001645The pebbling number of a connected graph.
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