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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000213
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Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000213: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 1 = 0 + 1
{{1,2}}
=> [2,1] => [1,2] => 2 = 1 + 1
{{1},{2}}
=> [1,2] => [2,1] => 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 3 = 2 + 1
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => 2 = 1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 4 = 3 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => 3 = 2 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => 3 = 2 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => 3 = 2 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => 3 = 2 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => 3 = 2 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => 3 = 2 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => 3 = 2 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => 3 = 2 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => 3 = 2 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => 3 = 2 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 5 = 4 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => 4 = 3 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => 4 = 3 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => 4 = 3 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => 4 = 3 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => 4 = 3 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => 3 = 2 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => 4 = 3 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => 4 = 3 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => 4 = 3 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => 4 = 3 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => 4 = 3 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => 4 = 3 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => 4 = 3 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => 2 = 1 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => 3 = 2 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => 4 = 3 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => 2 = 1 + 1
Description
The number of weak exceedances (also weak excedences) of a permutation.
This is defined as
$$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$$
The number of weak exceedances is given by the number of exceedances (see [[St000155]]) plus the number of fixed points (see [[St000022]]) of $\sigma$.
Matching statistic: St000245
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 1
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 0
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 2
{{1,2},{3}}
=> [2,1,3] => [3,2,1] => [2,3,1] => 1
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
=> [1,3,2] => [2,1,3] => [2,1,3] => 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [3,1,2] => 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 3
{{1,2,3},{4}}
=> [2,3,1,4] => [4,2,3,1] => [2,3,4,1] => 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
{{1,2},{3,4}}
=> [2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [3,2,4,1] => [2,4,1,3] => 2
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,2,4] => [1,3,2,4] => 2
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [4,3,2,1] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,3,2,1] => [3,2,4,1] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,3,2] => [1,3,4,2] => 2
{{1},{2,3,4}}
=> [1,3,4,2] => [2,1,3,4] => [2,1,3,4] => 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [2,4,3,1] => [3,4,1,2] => 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,4,2] => [1,4,2,3] => 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,3,1,4] => [3,1,2,4] => 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 2
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,2,3,4,1] => [2,3,4,5,1] => 3
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,2,3,1,5] => [2,3,4,1,5] => 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [4,2,3,5,1] => [2,3,5,1,4] => 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [5,2,1,3,4] => [2,5,4,3,1] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,2,4,3,1] => [2,4,3,5,1] => 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 3
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,2,1,4,5] => [2,3,1,4,5] => 3
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [3,2,5,4,1] => [2,4,5,1,3] => 3
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,4,5,3] => [1,2,5,3,4] => 3
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,2,1,5,4] => [2,3,1,5,4] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [3,2,4,1,5] => [2,4,1,3,5] => 3
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [3,2,4,5,1] => [2,5,1,3,4] => 3
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,2,4,3] => [4,5,3,2,1] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [5,3,2,4,1] => [3,2,4,5,1] => 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,5,2,3,4] => [1,5,4,3,2] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [4,1,2,3,5] => [4,3,2,1,5] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,5,2,3,1] => [5,1,4,3,2] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [4,1,2,5,3] => [5,3,2,1,4] => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,3,2,1,5] => [3,2,4,1,5] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [4,3,2,5,1] => [3,2,5,1,4] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,3,2,5] => [1,3,4,2,5] => 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,1,3,2,4] => [3,5,4,2,1] => 1
Description
The number of ascents of a permutation.
Matching statistic: St000250
St000250: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> ? = 0 + 2
{{1,2}}
=> 3 = 1 + 2
{{1},{2}}
=> 2 = 0 + 2
{{1,2,3}}
=> 4 = 2 + 2
{{1,2},{3}}
=> 3 = 1 + 2
{{1,3},{2}}
=> 3 = 1 + 2
{{1},{2,3}}
=> 3 = 1 + 2
{{1},{2},{3}}
=> 3 = 1 + 2
{{1,2,3,4}}
=> 5 = 3 + 2
{{1,2,3},{4}}
=> 4 = 2 + 2
{{1,2,4},{3}}
=> 4 = 2 + 2
{{1,2},{3,4}}
=> 4 = 2 + 2
{{1,2},{3},{4}}
=> 4 = 2 + 2
{{1,3,4},{2}}
=> 4 = 2 + 2
{{1,3},{2,4}}
=> 2 = 0 + 2
{{1,3},{2},{4}}
=> 3 = 1 + 2
{{1,4},{2,3}}
=> 4 = 2 + 2
{{1},{2,3,4}}
=> 4 = 2 + 2
{{1},{2,3},{4}}
=> 4 = 2 + 2
{{1,4},{2},{3}}
=> 4 = 2 + 2
{{1},{2,4},{3}}
=> 3 = 1 + 2
{{1},{2},{3,4}}
=> 4 = 2 + 2
{{1},{2},{3},{4}}
=> 4 = 2 + 2
{{1,2,3,4,5}}
=> 6 = 4 + 2
{{1,2,3,4},{5}}
=> 5 = 3 + 2
{{1,2,3,5},{4}}
=> 5 = 3 + 2
{{1,2,3},{4,5}}
=> 5 = 3 + 2
{{1,2,3},{4},{5}}
=> 5 = 3 + 2
{{1,2,4,5},{3}}
=> 5 = 3 + 2
{{1,2,4},{3,5}}
=> 3 = 1 + 2
{{1,2,4},{3},{5}}
=> 4 = 2 + 2
{{1,2,5},{3,4}}
=> 5 = 3 + 2
{{1,2},{3,4,5}}
=> 5 = 3 + 2
{{1,2},{3,4},{5}}
=> 5 = 3 + 2
{{1,2,5},{3},{4}}
=> 5 = 3 + 2
{{1,2},{3,5},{4}}
=> 4 = 2 + 2
{{1,2},{3},{4,5}}
=> 5 = 3 + 2
{{1,2},{3},{4},{5}}
=> 5 = 3 + 2
{{1,3,4,5},{2}}
=> 5 = 3 + 2
{{1,3,4},{2,5}}
=> 3 = 1 + 2
{{1,3,4},{2},{5}}
=> 4 = 2 + 2
{{1,3,5},{2,4}}
=> 3 = 1 + 2
{{1,3},{2,4,5}}
=> 3 = 1 + 2
{{1,3},{2,4},{5}}
=> 3 = 1 + 2
{{1,3,5},{2},{4}}
=> 4 = 2 + 2
{{1,3},{2,5},{4}}
=> 3 = 1 + 2
{{1,3},{2},{4,5}}
=> 4 = 2 + 2
{{1,3},{2},{4},{5}}
=> 4 = 2 + 2
{{1,4,5},{2,3}}
=> 5 = 3 + 2
{{1,4},{2,3,5}}
=> 3 = 1 + 2
{{1,4},{2,3},{5}}
=> 4 = 2 + 2
Description
The number of blocks ([[St000105]]) plus the number of antisingletons ([[St000248]]) of a set partition.
Matching statistic: St000702
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000702: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1] => ? = 0 + 1
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [2,1,3] => 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [3,2,1] => 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [3,1,2] => 2 = 1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 3 = 2 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [2,3,4,1] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 3 = 2 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [4,2,1,3] => 3 = 2 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [3,1,2,4] => 3 = 2 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [4,1,3,2] => 3 = 2 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [4,1,2,3] => 3 = 2 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 4 = 3 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,5,3,4] => 4 = 3 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 4 = 3 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 4 = 3 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,5,3,2,4] => 4 = 3 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [1,4,2,3,5] => 4 = 3 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,5,2,4,3] => 4 = 3 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,5,2,3,4] => 4 = 3 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => 4 = 3 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => [2,4,3,5,1] => 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => 3 = 2 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => [2,3,4,1,5] => 2 = 1 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => [2,3,5,4,1] => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => [2,3,5,1,4] => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => 3 = 2 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => [2,4,5,3,1] => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => [2,1,5,3,4] => 3 = 2 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => 4 = 3 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => [3,2,4,5,1] => 2 = 1 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => 3 = 2 + 1
Description
The number of weak deficiencies of a permutation.
This is defined as
$$\operatorname{wdec}(\sigma)=\#\{i:\sigma(i) \leq i\}.$$
The number of weak exceedances is [[St000213]], the number of deficiencies is [[St000703]].
Matching statistic: St000454
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> 0
{{1,2,3}}
=> [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,3,2] => ([(1,2)],3)
=> 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,2,2,2}
{{1,2,3},{4}}
=> [2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,2,2,2,2}
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,2,2,2}
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,2,2,2}
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,2,2,2}
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3}
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [2,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [2,6,5,1,4,3] => ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,6,5,4,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3},{4},{5,6}}
=> [2,3,1,4,6,5] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [2,6,1,5,4,3] => ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4}
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,2},{3,4,5},{6}}
=> [2,1,4,5,3,6] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,2},{3,4,6},{5}}
=> [2,1,4,6,5,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,2},{3,4},{5},{6}}
=> [2,1,4,3,5,6] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,2},{3,5,6},{4}}
=> [2,1,5,4,6,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,2},{3,5},{4},{6}}
=> [2,1,5,4,3,6] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
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.
Matching statistic: St000259
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2] => [2] => ([],2)
=> ? = 0
{{1},{2}}
=> [1,1] => [1,1] => ([(0,1)],2)
=> 1
{{1,2,3}}
=> [3] => [3] => ([],3)
=> ? ∊ {1,1,1}
{{1,2},{3}}
=> [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1}
{{1,3},{2}}
=> [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1}
{{1},{2,3}}
=> [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
{{1,2,3,4}}
=> [4] => [4] => ([],4)
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
{{1,2,3},{4}}
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
{{1,2,4},{3}}
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
{{1,2},{3,4}}
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
{{1,3,4},{2}}
=> [3,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
{{1,3},{2,4}}
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
{{1,4},{2,3}}
=> [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
{{1},{2,3,4}}
=> [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,1,2,2,2,2,2,2,2,3}
{{1},{2,4},{3}}
=> [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
{{1,2,3,4,5}}
=> [5] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2},{3,4},{5}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2},{3,5},{4}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3},{2,4},{5}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3},{2,5},{4}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,4},{2,3},{5}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,5},{2,3,4}}
=> [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1},{2,3,4,5}}
=> [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
{{1},{2,3,4},{5}}
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1,5},{2,3},{4}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1},{2,3,5},{4}}
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2,3},{4,5}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1,4,5},{2},{3}}
=> [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,4},{2,5},{3}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,5},{2,4},{3}}
=> [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1},{2,4,5},{3}}
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2,4},{3,5}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1,5},{2},{3,4}}
=> [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1},{2,5},{3,4}}
=> [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2},{3,4,5}}
=> [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
{{1},{2,3,4,5,6}}
=> [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
{{1},{2,3,4,5},{6}}
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3,4,6},{5}}
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3,4},{5,6}}
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3,5,6},{4}}
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3,5},{4,6}}
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3,5},{4},{6}}
=> [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3,6},{4,5}}
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3},{4,5,6}}
=> [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3},{4,5},{6}}
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3,6},{4},{5}}
=> [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3},{4,6},{5}}
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3},{4},{5,6}}
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,3},{4},{5},{6}}
=> [1,2,1,1,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,4,5,6},{3}}
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,4,5},{3,6}}
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,4,5},{3},{6}}
=> [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,4,6},{3,5}}
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,4},{3,5,6}}
=> [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,4},{3,5},{6}}
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,4,6},{3},{5}}
=> [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,4},{3,6},{5}}
=> [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,4},{3},{5,6}}
=> [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,4},{3},{5},{6}}
=> [1,2,1,1,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1},{2,5,6},{3,4}}
=> [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001060
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 50%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> [1] => [1] => ([],1)
=> ? = 0
{{1,2}}
=> [2,1] => [1,2] => ([],2)
=> ? ∊ {0,1}
{{1},{2}}
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {0,1}
{{1,2,3}}
=> [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2}
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2}
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,2}
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2}
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,1,1,2}
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3}
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4}
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,3,6},{2,4},{5}}
=> [3,4,6,2,5,1] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,3,5,6},{2},{4}}
=> [3,2,5,4,6,1] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,3,5},{2,6},{4}}
=> [3,6,5,4,1,2] => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
{{1,3,6},{2},{4,5}}
=> [3,2,6,5,4,1] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,3,6},{2},{4},{5}}
=> [3,2,6,4,5,1] => [1,3,6,2,4,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,4,6},{2,3,5}}
=> [4,3,5,6,2,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,4,6},{2,3},{5}}
=> [4,3,2,6,5,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,5,6},{2,3,4}}
=> [5,3,4,2,6,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
{{1,6},{2,3,4},{5}}
=> [6,3,4,2,5,1] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
{{1,5,6},{2,3},{4}}
=> [5,3,2,4,6,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,5},{2,3,6},{4}}
=> [5,3,6,4,1,2] => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,6},{2,3,5},{4}}
=> [6,3,5,4,2,1] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,6},{2,3},{4,5}}
=> [6,3,2,5,4,1] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
{{1,6},{2,3},{4},{5}}
=> [6,3,2,4,5,1] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
{{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
{{1,4},{2,5,6},{3}}
=> [4,5,3,1,6,2] => [1,4,2,5,6,3] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> 2
{{1,4,6},{2},{3,5}}
=> [4,2,5,6,3,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,4},{2,6},{3,5}}
=> [4,6,5,1,3,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
{{1,4,6},{2},{3},{5}}
=> [4,2,3,6,5,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
{{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
{{1,5,6},{2,4},{3}}
=> [5,4,3,2,6,1] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
{{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
{{1,6},{2,4,5},{3}}
=> [6,4,3,5,2,1] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,6},{2,4},{3},{5}}
=> [6,4,3,2,5,1] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
{{1,5,6},{2},{3,4}}
=> [5,2,4,3,6,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001863
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 83%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001863: Signed permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 83%
Values
{{1}}
=> [1] => [1] => [1] => 1 = 0 + 1
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
{{1,2},{3}}
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
{{1,3},{2}}
=> [3,2,1] => [2,1,3] => [2,1,3] => 2 = 1 + 1
{{1},{2,3}}
=> [1,3,2] => [3,2,1] => [3,2,1] => 2 = 1 + 1
{{1},{2},{3}}
=> [1,2,3] => [2,3,1] => [2,3,1] => 2 = 1 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 3 = 2 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 1 = 0 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 3 = 2 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 3 = 2 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,1,4] => [2,3,1,4] => 3 = 2 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,4,3,1] => [2,4,3,1] => 3 = 2 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [2,3,4,1] => [2,3,4,1] => 3 = 2 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 4 = 3 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => 4 = 3 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 4 = 3 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 4 = 3 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => 4 = 3 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,3,4,2,5] => [1,3,4,2,5] => 4 = 3 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => 4 = 3 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => 4 = 3 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,1,3,2,4] => [5,1,3,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [5,1,2,4,3] => [5,1,2,4,3] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [4,1,2,5,3] => [4,1,2,5,3] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [5,1,4,2,3] => [5,1,4,2,3] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,1,4,5,3] => [2,1,4,5,3] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,1,4,5] => [3,2,1,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [5,2,1,3,4] => [5,2,1,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,1,5,4] => [3,2,1,5,4] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,2,3,1,5] => [4,2,3,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,1,5] => [3,2,4,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [3,2,5,4,1] => [3,2,5,4,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,1,4,5] => [2,3,1,4,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [5,3,1,2,4] => [5,3,1,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,5,1,3,4] => [2,5,1,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,3,1,5,4] => [2,3,1,5,4] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [4,3,2,1,5] => [4,3,2,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,1,5] => [2,4,3,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,1,5] => [2,3,4,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [5,3,4,2,1] => [5,3,4,2,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [2,3,5,4,1] => [2,3,5,4,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,2,3,5,6,4] => [1,2,3,5,6,4] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,2,5,4,6,3] => [1,2,5,4,6,3] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,2,4,5,3,6] => [1,2,4,5,3,6] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5} + 1
Description
The number of weak excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) \geq i\}\rvert$.
Matching statistic: St001626
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001626: Lattices ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001626: Lattices ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => ([],1)
=> ([(0,1)],2)
=> 2 = 0 + 2
{{1,2}}
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
{{1},{2}}
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
{{1,2,3}}
=> [2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3 = 1 + 2
{{1,2},{3}}
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 1 + 2
{{1,3},{2}}
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 1 + 2
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 1 + 2
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
{{1,2,3,4}}
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {0,1,1,2,2,2,2,2} + 2
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 2 + 2
{{1,2,4},{3}}
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,1,1,2,2,2,2,2} + 2
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 4 = 2 + 2
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 2 + 2
{{1,3,4},{2}}
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? ∊ {0,1,1,2,2,2,2,2} + 2
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {0,1,1,2,2,2,2,2} + 2
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? ∊ {0,1,1,2,2,2,2,2} + 2
{{1,4},{2,3}}
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? ∊ {0,1,1,2,2,2,2,2} + 2
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 4 = 2 + 2
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4 = 2 + 2
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? ∊ {0,1,1,2,2,2,2,2} + 2
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,1,1,2,2,2,2,2} + 2
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 4 = 2 + 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 3 + 2
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> ([(0,2),(0,3),(1,11),(2,1),(2,12),(3,4),(3,5),(3,12),(4,8),(4,10),(5,8),(5,9),(6,14),(7,14),(8,13),(9,6),(9,13),(10,7),(10,13),(11,6),(11,7),(12,9),(12,10),(12,11),(13,14)],15)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,3),(0,4),(1,11),(2,10),(3,2),(3,9),(4,1),(4,9),(5,7),(5,8),(6,12),(7,12),(8,12),(9,5),(9,10),(9,11),(10,6),(10,7),(11,6),(11,8)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,10),(2,7),(2,8),(3,9),(3,12),(4,9),(4,11),(5,2),(5,11),(5,12),(7,14),(8,14),(9,1),(9,13),(10,6),(11,7),(11,13),(12,8),(12,13),(13,10),(13,14),(14,6)],15)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => ([(3,4)],5)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 5 = 3 + 2
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,8),(1,10),(2,7),(2,9),(3,11),(3,12),(4,2),(4,11),(4,13),(5,1),(5,12),(5,13),(6,17),(7,15),(8,16),(9,6),(9,15),(10,6),(10,16),(11,7),(11,14),(12,8),(12,14),(13,9),(13,10),(13,14),(14,15),(14,16),(15,17),(16,17)],18)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,11),(2,10),(3,8),(3,9),(4,7),(4,8),(5,7),(5,9),(7,12),(8,2),(8,12),(9,1),(9,12),(10,6),(11,6),(12,10),(12,11)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => ([],5)
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(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)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5 = 3 + 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 5 = 3 + 2
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
Description
The number of maximal proper sublattices of a lattice.
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