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Your data matches 122 different statistics following compositions of up to 3 maps.
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Matching statistic: St001714
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Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => [2,1]
=> 0
[2,1,3] => [1,3,2] => [2,1]
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1]
=> 0
[1,3,2,4] => [1,3,2,4] => [3,1]
=> 0
[1,3,4,2] => [1,3,4,2] => [3,1]
=> 0
[1,4,2,3] => [1,4,2,3] => [3,1]
=> 0
[1,4,3,2] => [1,4,2,3] => [3,1]
=> 0
[2,1,3,4] => [1,3,4,2] => [3,1]
=> 0
[2,1,4,3] => [1,4,2,3] => [3,1]
=> 0
[2,3,1,4] => [1,4,2,3] => [3,1]
=> 0
[2,4,1,3] => [1,3,2,4] => [3,1]
=> 0
[2,4,3,1] => [1,2,4,3] => [3,1]
=> 0
[3,1,2,4] => [1,2,4,3] => [3,1]
=> 0
[3,1,4,2] => [1,4,2,3] => [3,1]
=> 0
[3,2,1,4] => [1,4,2,3] => [3,1]
=> 0
[3,2,4,1] => [1,2,4,3] => [3,1]
=> 0
[4,1,3,2] => [1,3,2,4] => [3,1]
=> 0
[4,2,1,3] => [1,3,2,4] => [3,1]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> 0
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> 0
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> 0
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> 0
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> 0
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> 0
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> 0
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> 0
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> 0
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> 0
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
=> 0
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
=> 0
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
=> 0
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
=> 0
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
=> 0
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
=> 0
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> 0
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
=> 0
[2,1,5,4,3] => [1,5,2,3,4] => [4,1]
=> 0
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> 0
[2,3,1,5,4] => [1,5,2,3,4] => [4,1]
=> 0
[2,3,4,1,5] => [1,5,2,3,4] => [4,1]
=> 0
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St001222
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001222: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 67%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001222: Dyck paths ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 67%
Values
[1,3,2] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,5,4,3] => [1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1,5,4] => [1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,4,1,5] => [1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,4,6,7,5] => [1,2,3,4,6,7,5] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,4,7,6,5] => [1,2,3,4,7,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,5,6,4,7] => [1,2,3,5,6,4,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,5,6,7,4] => [1,2,3,5,6,7,4] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,7,4,6,5] => [1,2,3,7,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,3,7,5,4,6] => [1,2,3,7,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,3,7,5,6,4] => [1,2,3,7,4,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,7,6,4,5] => [1,2,3,7,4,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,3,7,6,5,4] => [1,2,3,7,4,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,4,5,6,3,7] => [1,2,4,5,6,3,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,4,5,6,7,3] => [1,2,4,5,6,7,3] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,5,4,6,7,3] => [1,2,5,3,4,6,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,6,3,5,7,4] => [1,2,6,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,6,4,3,5,7] => [1,2,6,3,5,7,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,6,4,5,7,3] => [1,2,6,3,4,5,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,6,4,7,3,5] => [1,2,6,3,5,4,7] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,6,5,7,3,4] => [1,2,6,3,4,5,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,6,5,7,4,3] => [1,2,6,3,4,5,7] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,3,5,6,4] => [1,2,7,3,5,6,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,3,6,5,4] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,4,3,5,6] => [1,2,7,3,5,6,4] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,4,3,6,5] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,4,5,3,6] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,4,5,6,3] => [1,2,7,3,4,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,7,4,6,3,5] => [1,2,7,3,5,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,4,6,5,3] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,5,3,4,6] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,5,3,6,4] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,5,4,3,6] => [1,2,7,3,6,4,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,5,4,6,3] => [1,2,7,3,4,6,5] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
[1,2,7,5,6,3,4] => [1,2,7,3,4,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,7,5,6,4,3] => [1,2,7,3,4,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,7,6,3,4,5] => [1,2,7,3,4,5,6] => [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0 + 1
[1,2,7,6,3,5,4] => [1,2,7,3,5,4,6] => [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 + 1
Description
Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module.
Matching statistic: St000366
Mp00223: Permutations —runsort⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 67%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 67%
Values
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> [3,1,2] => 0
[2,1,3] => [1,3,2] => [[1,2],[3]]
=> [3,1,2] => 0
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[1,3,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,4,2,3] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[1,4,3,2] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[2,1,3,4] => [1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => 0
[2,1,4,3] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[2,3,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[2,4,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,1,2,4] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 0
[3,1,4,2] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[3,2,1,4] => [1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[3,2,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 0
[4,1,3,2] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 0
[1,2,5,4,3] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[1,3,5,4,2] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[1,4,2,5,3] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 0
[1,4,3,2,5] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 0
[1,4,3,5,2] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[1,4,5,2,3] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[1,4,5,3,2] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[1,5,2,3,4] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[1,5,2,4,3] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1
[1,5,3,2,4] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1
[1,5,3,4,2] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[2,1,3,4,5] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[2,1,3,5,4] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[2,1,4,3,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[2,1,4,5,3] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[2,1,5,3,4] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[2,3,1,4,5] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[2,3,1,5,4] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,2,3,6,4,7,5] => [1,2,3,6,4,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,2,3,6,5,4,7] => [1,2,3,6,4,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,2,4,3,6,7,5] => [1,2,4,3,6,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,2,5,3,6,7,4] => [1,2,5,3,6,7,4] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,2,5,4,3,6,7] => [1,2,5,3,6,7,4] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,2,5,4,7,6,3] => [1,2,5,3,4,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,2,5,6,3,7,4] => [1,2,5,6,3,7,4] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,2,5,6,4,3,7] => [1,2,5,6,3,7,4] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 0
[1,2,6,3,4,7,5] => [1,2,6,3,4,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,2,6,4,7,3,5] => [1,2,6,3,5,4,7] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,2,6,4,7,5,3] => [1,2,6,3,4,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,2,6,5,3,4,7] => [1,2,6,3,4,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,2,6,5,4,7,3] => [1,2,6,3,4,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
[1,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,2,7,3,6,5,4] => [1,2,7,3,6,4,5] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,2,7,4,3,6,5] => [1,2,7,3,6,4,5] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,2,7,4,5,3,6] => [1,2,7,3,6,4,5] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,2,7,4,6,3,5] => [1,2,7,3,5,4,6] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,2,7,5,3,6,4] => [1,2,7,3,6,4,5] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,2,7,5,4,3,6] => [1,2,7,3,6,4,5] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,2,7,6,3,5,4] => [1,2,7,3,5,4,6] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,2,7,6,4,3,5] => [1,2,7,3,5,4,6] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 0
[1,3,2,4,6,7,5] => [1,3,2,4,6,7,5] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 0
[1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => ? = 0
[1,3,2,5,6,7,4] => [1,3,2,5,6,7,4] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 0
[1,3,2,5,7,4,6] => [1,3,2,5,7,4,6] => [[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => ? = 0
[1,3,2,5,7,6,4] => [1,3,2,5,7,4,6] => [[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => ? = 0
[1,3,2,6,4,5,7] => [1,3,2,6,4,5,7] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,3,2,6,4,7,5] => [1,3,2,6,4,7,5] => [[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => ? = 0
[1,3,2,6,5,4,7] => [1,3,2,6,4,7,5] => [[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => ? = 0
[1,3,2,6,5,7,4] => [1,3,2,6,4,5,7] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,3,2,6,7,4,5] => [1,3,2,6,7,4,5] => [[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => ? = 0
[1,3,2,6,7,5,4] => [1,3,2,6,7,4,5] => [[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => ? = 0
[1,3,2,7,4,5,6] => [1,3,2,7,4,5,6] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,3,2,7,4,6,5] => [1,3,2,7,4,6,5] => [[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => ? = 0
[1,3,2,7,5,4,6] => [1,3,2,7,4,6,5] => [[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => ? = 0
[1,3,2,7,5,6,4] => [1,3,2,7,4,5,6] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,3,2,7,6,4,5] => [1,3,2,7,4,5,6] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,3,2,7,6,5,4] => [1,3,2,7,4,5,6] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 0
[1,3,4,2,5,7,6] => [1,3,4,2,5,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 0
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000455
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 33%
Values
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[2,1,3] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,3,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,4,3,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,1,3,4] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,1,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,3,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,4,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,4,3,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,1,2,4] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,1,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,2,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[4,1,3,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[4,2,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,5,4,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,3,4,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,4,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,4,3,5,2] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,5,3,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,5,2,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,5,2,4,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,5,3,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,5,3,4,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,5,4,2,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[1,5,4,3,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,1,3,4,5] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,1,3,5,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,1,4,3,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,1,4,5,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,1,5,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,1,5,4,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,3,1,4,5] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,3,1,5,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,3,4,1,5] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,3,5,1,4] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,3,5,4,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,4,1,3,5] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,4,1,5,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,4,3,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,4,3,5,1] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,4,5,1,3] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,5,1,4,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,5,3,1,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,5,4,1,3] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,1,4,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[3,1,5,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,2,4,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,2,5,1,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[4,1,3,2,5] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[4,2,5,1,3] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001330
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[1,3,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[2,1,3] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,3,4,2] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,1,3,4] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,3,4,2,5] => [1,3,4,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,3,4,5,2] => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,2,4] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,2,5,3] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,4,3,2,5] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,2,4,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,3,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[2,1,3,5,4] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,4,5,3] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,1,4,5] => [1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,5,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,1,3,5] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,1,5,3] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,4,3,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,4,3,5,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,5,1,3] => [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,5,1,4,3] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,5,3,1,4] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 2
[2,5,4,1,3] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[3,1,4,2,5] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 2
[3,1,5,2,4] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,2,4,1,5] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[3,2,5,1,4] => [1,4,2,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 2
[4,1,3,2,5] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[4,2,5,1,3] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [5,3,1,2,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [6,3,1,4,2,5] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [5,3,1,6,2,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [5,3,1,6,2,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [6,3,1,2,5,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [6,3,1,2,5,4] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,4,2,3,6,5] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,4,3,6,5,2] => [1,4,2,3,6,5] => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,4,5,2,6,3] => [1,4,5,2,6,3] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,4,5,3,2,6] => [1,4,5,2,6,3] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,4,6,2,5,3] => [1,4,6,2,5,3] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,4,6,3,2,5] => [1,4,6,2,5,3] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [5,1,4,2,3,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,5,2,4,6,3] => [1,5,2,4,6,3] => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000454
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 2
[2,1,3] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 2
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[1,3,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,4,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[1,4,3,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[2,1,3,4] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[2,1,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[2,3,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[2,4,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[2,4,3,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[3,1,2,4] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[3,1,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[3,2,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[3,2,4,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 2
[4,1,3,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[4,2,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,2,4,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,5,3,4] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,2,5,4,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,3,4,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,3,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,3,5,2,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,4,3,5,2] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,5,2,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,5,2,4,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,3,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,5,3,4,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,5,4,2,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,5,4,3,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[2,1,3,4,5] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,1,3,5,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,1,4,3,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[2,1,4,5,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,1,5,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[2,1,5,4,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[2,3,1,4,5] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,3,1,5,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[2,3,4,1,5] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[2,3,5,1,4] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[2,3,5,4,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[2,4,1,3,5] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[2,4,1,5,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,4,3,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[2,4,5,3,1] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,1,2,4,5] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,2,4,5,1] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,5,4,1,2] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[3,5,4,2,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,1,2,3,5] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,2,3,5,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,3,5,1,2] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[4,3,5,2,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,1,3,4,6,5] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,1,3,5,6,4] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,1,4,5,6,3] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,3,1,4,5,6] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,4,1,3,5,6] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,4,5,3,6,1] => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,4,6,3,5,1] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,4,6,5,3,1] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,5,1,3,4,6] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,5,6,3,4,1] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,5,6,4,3,1] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[2,6,1,3,4,5] => [1,3,4,5,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[3,1,2,4,6,5] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[3,1,2,5,6,4] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[3,1,4,5,6,2] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[3,2,1,4,5,6] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[3,2,4,6,5,1] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[3,2,5,6,4,1] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000422
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 4
[2,1,3] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 4
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[1,3,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[1,4,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[1,4,3,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[2,1,3,4] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[2,1,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[2,3,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[2,4,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[2,4,3,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[3,1,2,4] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[3,1,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[3,2,1,4] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[3,2,4,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 4
[4,1,3,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[4,2,1,3] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 4
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,2,4,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,2,5,3,4] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,2,5,4,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,3,4,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,3,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,3,5,2,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,3,5,4,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,4,2,5,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,4,3,2,5] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,4,3,5,2] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,4,5,3,2] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[1,5,2,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,5,2,4,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[1,5,3,2,4] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[1,5,3,4,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,5,4,2,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[1,5,4,3,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[2,1,3,4,5] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[2,1,3,5,4] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[2,1,4,3,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[2,1,4,5,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[2,1,5,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[2,1,5,4,3] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[2,3,1,4,5] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[2,3,1,5,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[2,3,4,1,5] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[2,3,5,1,4] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 4
[2,3,5,4,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[2,4,1,3,5] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 4
[2,4,1,5,3] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,4,3,1,5] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 4
[2,4,5,3,1] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[3,1,2,4,5] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[3,2,4,5,1] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[3,5,4,1,2] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[3,5,4,2,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[4,1,2,3,5] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[4,2,3,5,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[4,3,5,1,2] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[4,3,5,2,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,3,6,4,5] => [1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,3,6,5,4] => [1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,4,5,3,6] => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,3,4,5,2,6] => [1,3,4,5,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,3,5,6,4,2] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,1,3,4,6,5] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,1,3,5,6,4] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,1,4,5,6,3] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,3,1,4,5,6] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,3,5,4,6,1] => [1,2,3,5,4,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,3,6,4,5,1] => [1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,3,6,5,4,1] => [1,2,3,6,4,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,4,1,3,5,6] => [1,3,5,6,2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,4,5,3,6,1] => [1,2,4,5,3,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,4,6,3,5,1] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,4,6,5,3,1] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,5,1,3,4,6] => [1,3,4,6,2,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,5,6,3,4,1] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,5,6,4,3,1] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[2,6,1,3,4,5] => [1,3,4,5,2,6] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[3,1,2,4,6,5] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[3,1,2,5,6,4] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[3,1,4,5,6,2] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[3,2,1,4,5,6] => [1,4,5,6,2,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[3,2,4,6,5,1] => [1,2,4,6,3,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[3,2,5,6,4,1] => [1,2,5,6,3,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000759
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,3,2] => [2,3,1] => ([(1,2)],3)
=> [3]
=> 1 = 0 + 1
[2,1,3] => [3,1,2] => ([(1,2)],3)
=> [3]
=> 1 = 0 + 1
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 1 = 0 + 1
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 1 = 0 + 1
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 1 = 0 + 1
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 1 = 0 + 1
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> [4,4,4]
=> 1 = 0 + 1
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> [4,2]
=> 1 = 0 + 1
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> [8]
=> 1 = 0 + 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 1 = 0 + 1
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 1 = 0 + 1
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> [8]
=> 1 = 0 + 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 1 = 0 + 1
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> [4]
=> 1 = 0 + 1
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 1 = 0 + 1
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1 = 0 + 1
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1 = 0 + 1
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0 + 1
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0 + 1
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 0 + 1
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 0 + 1
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 0 + 1
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0 + 1
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 0 + 1
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 0 + 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 0 + 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0 + 1
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 0 + 1
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 0 + 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0 + 1
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 0 + 1
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 0 + 1
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 0 + 1
[1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 1 = 0 + 1
[1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 0 + 1
[1,5,2,4,3] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 + 1
[1,5,3,2,4] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1 + 1
[1,5,3,4,2] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 1 = 0 + 1
[1,5,4,2,3] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 1 = 0 + 1
[1,5,4,3,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 1 = 0 + 1
[2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0 + 1
[2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 0 + 1
[2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 0 + 1
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 0 + 1
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 0 + 1
[2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 1 = 0 + 1
[2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 0 + 1
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 0 + 1
[2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 0 + 1
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 0 + 1
[2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 1 = 0 + 1
[2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0 + 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> ? = 1 + 1
[2,4,3,1,5] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 + 1
[2,4,3,5,1] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 1 = 0 + 1
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 0 + 1
[2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 1 = 0 + 1
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 0 + 1
[2,5,1,4,3] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[2,5,3,1,4] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 1 = 0 + 1
[2,5,3,4,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 1 = 0 + 1
[2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 1 = 0 + 1
[2,5,4,3,1] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 1 = 0 + 1
[3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 0 + 1
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 0 + 1
[3,1,4,2,5] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0 + 1
[3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 0 + 1
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> ? = 1 + 1
[3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 1 = 0 + 1
[3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 0 + 1
[3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 1 = 0 + 1
[3,2,4,1,5] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1 + 1
[3,2,4,5,1] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 1 = 0 + 1
[3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 1 = 0 + 1
[3,2,5,4,1] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> 1 = 0 + 1
[3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 0 + 1
[3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 0 + 1
[3,4,2,1,5] => [5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 1 = 0 + 1
[3,4,2,5,1] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 1 = 0 + 1
[3,5,1,2,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 0 + 1
[3,5,1,4,2] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 1 = 0 + 1
[3,5,2,1,4] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 1 = 0 + 1
[3,5,2,4,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 1 = 0 + 1
[3,5,4,1,2] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 1 = 0 + 1
[3,5,4,2,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 1 = 0 + 1
[4,1,2,3,5] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 0 + 1
[4,1,2,5,3] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 0 + 1
[4,1,3,2,5] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 0 + 1
[4,1,3,5,2] => [2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 1 = 0 + 1
[4,1,5,2,3] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 0 + 1
[4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 1 = 0 + 1
[4,2,1,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 0 + 1
[4,2,1,5,3] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[4,2,3,1,5] => [5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 1 = 0 + 1
[4,2,3,5,1] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 1 = 0 + 1
[4,2,5,1,3] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 1 = 0 + 1
[4,2,5,3,1] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 1 = 0 + 1
[1,2,3,4,6,5] => [5,6,4,3,2,1] => ([(4,5)],6)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 0 + 1
[1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(4,5)],6)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 0 + 1
Description
The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St000475
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,3,2] => [2,3,1] => ([(1,2)],3)
=> [3]
=> 0
[2,1,3] => [3,1,2] => ([(1,2)],3)
=> [3]
=> 0
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 0
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 0
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 0
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 0
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 0
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> [4,4,4]
=> 0
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> [4,2]
=> 0
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> [8]
=> 0
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 0
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> [8]
=> 0
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 0
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> [4]
=> 0
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 0
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 0
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 0
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 0
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 0
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 0
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 0
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 0
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 0
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 0
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 0
[1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0
[1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 0
[1,5,2,4,3] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1
[1,5,3,2,4] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1
[1,5,3,4,2] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 0
[1,5,4,2,3] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0
[1,5,4,3,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 0
[2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0
[2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 0
[2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 0
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 0
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 0
[2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 0
[2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 0
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 0
[2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 0
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 0
[2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0
[2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> ? = 1
[2,4,3,1,5] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1
[2,4,3,5,1] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 0
[2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 0
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 0
[2,5,1,4,3] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0
[2,5,3,1,4] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 0
[2,5,3,4,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 0
[2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0
[2,5,4,3,1] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 0
[3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 0
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 0
[3,1,4,2,5] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0
[3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 0
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> ? = 1
[3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0
[3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 0
[3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 0
[3,2,4,1,5] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1
[3,2,4,5,1] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0
[3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0
[3,2,5,4,1] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> 0
[3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 0
[3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 0
[3,4,2,1,5] => [5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0
[3,4,2,5,1] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 0
[3,5,1,2,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 0
[3,5,1,4,2] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 0
[3,5,2,1,4] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0
[3,5,2,4,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 0
[3,5,4,1,2] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 0
[3,5,4,2,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0
[4,1,2,3,5] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 0
[4,1,2,5,3] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 0
[4,1,3,2,5] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 0
[4,1,3,5,2] => [2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 0
[4,1,5,2,3] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 0
[4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0
[4,2,1,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 0
[4,2,1,5,3] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0
[4,2,3,1,5] => [5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 0
[4,2,3,5,1] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 0
[4,2,5,1,3] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 0
[4,2,5,3,1] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 0
[1,2,3,4,6,5] => [5,6,4,3,2,1] => ([(4,5)],6)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 0
[1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(4,5)],6)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000929
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,3,2] => [2,3,1] => ([(1,2)],3)
=> [3]
=> 0
[2,1,3] => [3,1,2] => ([(1,2)],3)
=> [3]
=> 0
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> [4,4,4]
=> 0
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> [4,4,4]
=> 0
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> [8]
=> 0
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> [8]
=> 0
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> [4]
=> 0
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
=> [4,4,4]
=> 0
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> [4,2]
=> 0
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> [8]
=> 0
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 0
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> [8]
=> 0
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 0
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> [4]
=> 0
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> [3]
=> 0
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 0
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 0
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 0
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 0
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 0
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 0
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 0
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 0
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 0
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 0
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 0
[1,4,5,3,2] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0
[1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 0
[1,5,2,4,3] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1
[1,5,3,2,4] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 1
[1,5,3,4,2] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 0
[1,5,4,2,3] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 0
[1,5,4,3,2] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> [5]
=> 0
[2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5)
=> [5,5,5,5,5,5,5,5,5,5,5,5]
=> ? = 0
[2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 0
[2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 0
[2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 0
[2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 0
[2,1,5,4,3] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 0
[2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> [10,10,10,10]
=> ? = 0
[2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 0
[2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> ? = 0
[2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 0
[2,3,5,4,1] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0
[2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> ? = 1
[2,4,3,1,5] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1
[2,4,3,5,1] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0
[2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 0
[2,4,5,3,1] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 0
[2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 0
[2,5,1,4,3] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0
[2,5,3,1,4] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 0
[2,5,3,4,1] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 0
[2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0
[2,5,4,3,1] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 0
[3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> [10,10,10,10]
=> ? = 0
[3,1,2,5,4] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 0
[3,1,4,2,5] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 0
[3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 0
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> ? = 1
[3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0
[3,2,1,4,5] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> [5,5,5,5]
=> ? = 0
[3,2,1,5,4] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 0
[3,2,4,1,5] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> [15]
=> ? = 1
[3,2,4,5,1] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 0
[3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0
[3,2,5,4,1] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> 0
[3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> ? = 0
[3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 0
[3,4,2,1,5] => [5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 0
[3,4,2,5,1] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 0
[3,5,1,2,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 0
[3,5,1,4,2] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 0
[3,5,2,1,4] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0
[3,5,2,4,1] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 0
[3,5,4,1,2] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 0
[3,5,4,2,1] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> [3]
=> 0
[4,1,2,3,5] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> ? = 0
[4,1,2,5,3] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 0
[4,1,3,2,5] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 0
[4,1,3,5,2] => [2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 0
[4,1,5,2,3] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 0
[4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0
[4,2,1,3,5] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 0
[4,2,1,5,3] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0
[4,2,3,1,5] => [5,1,3,2,4] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 0
[4,2,3,5,1] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 0
[4,2,5,1,3] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 0
[4,2,5,3,1] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 0
[1,2,3,4,6,5] => [5,6,4,3,2,1] => ([(4,5)],6)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 0
[1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(4,5)],6)
=> [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]
=> ? = 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
The following 112 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001568The smallest positive integer that does not appear twice in the partition. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001651The Frankl number of a lattice. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St000068The number of minimal elements in a poset. St001866The nesting alignments of a signed permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001618The cardinality of the Frattini sublattice of a lattice. 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 $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001896The number of right descents of a signed permutations. St001893The flag descent of a signed permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001851The number of Hecke atoms of a signed permutation. St000312The number of leaves in a graph. St001889The size of the connectivity set of a signed permutation. St001625The Möbius invariant of a lattice. St001429The number of negative entries in a signed permutation. St000850The number of 1/2-balanced pairs in a poset. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001964The interval resolution global dimension of a poset. St000627The exponent of a binary word. St000878The number of ones minus the number of zeros of a binary word. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St000181The number of connected components of the Hasse diagram for the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001890The maximum magnitude of the Möbius function of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001490The number of connected components of a skew partition. St001877Number of indecomposable injective modules with projective dimension 2. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001754The number of tolerances of a finite lattice. St000098The chromatic number of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St000635The number of strictly order preserving maps of a poset into itself. St000741The Colin de Verdière graph invariant. St001333The cardinality of a minimal edge-isolating set of a graph.
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