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Your data matches 20 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> []
=> 0
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> []
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> []
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1]
=> 1
Description
The largest part of an integer partition.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 0
[1,0,1,0]
=> [1,2] => [2]
=> 1
[1,1,0,0]
=> [2,1] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,2,4,3,6,8,7,5] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,4,8,6,7,3] => ?
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,2,6,4,5,3,7,8] => ?
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,2,6,4,5,8,7,3] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,6,5,4,8,7,3] => ?
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,2,8,5,4,7,6,3] => ?
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,3,2,4,8,6,7,5] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,5,7,6,8,4] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,3,2,6,5,7,4,8] => ?
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,8,7,6] => ?
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6,8,7] => ?
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,4,6,5,7,2,8] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,4,7,6,5,8,2] => ?
=> ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,4,8,6,5,7,2] => ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,3,5,4,2,6,7,8] => ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,3,5,4,6,2,7,8] => ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,3,5,4,6,7,2,8] => ?
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,3,5,4,6,7,8,2] => ?
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,3,6,4,5,7,2,8] => ?
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,3,6,4,5,8,7,2] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,3,6,5,4,8,7,2] => ?
=> ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,3,7,6,5,4,2,8] => ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,3,5,2,7,6,8] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,3,5,2,7,8,6] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,3,5,2,8,7,6] => ?
=> ? = 3
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,4,3,5,8,7,6,2] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,3,6,5,7,2,8] => ?
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,5,3,4,6,2,7,8] => ?
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,5,3,4,7,6,2,8] => ?
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,5,3,4,8,6,7,2] => ?
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,8,3,5,4,6,7,2] => ?
=> ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,7,4,5,6,3,2,8] => ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,6,5,4,3,7,8,2] => ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,7,5,4,3,6,2,8] => ?
=> ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,7,5,4,3,6,8,2] => ?
=> ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,8,6,4,5,3,7,2] => ?
=> ? = 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,4,6,7,5,8] => ?
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4,6,8,7] => ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,3,5,6,8,7] => ?
=> ? = 2
[1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,1,5,8,6,7,4] => ?
=> ? = 2
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,3,1,6,5,7,4,8] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,4,6,5,1,7,8] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,4,1,7,6,8] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,3,5,4,6,1,7,8] => ?
=> ? = 1
[1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [2,3,5,4,8,7,6,1] => ?
=> ? = 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,6,4,5,7,8,1] => ?
=> ? = 1
[1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [2,4,3,1,8,6,7,5] => ?
=> ? = 3
[1,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [2,4,3,5,1,6,8,7] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,4,3,5,1,7,6,8] => ?
=> ? = 2
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,7,6,8,1] => ?
=> ? = 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000251
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 67%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [[1]]
=> {{1}}
=> ? = 0
[1,0,1,0]
=> [1,2] => [[1,2]]
=> {{1,2}}
=> 1
[1,1,0,0]
=> [2,1] => [[1],[2]]
=> {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> {{1,2,3}}
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> {{1,2},{3}}
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> {{1,2,3,4}}
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,4,6,8,7,5] => [[1,2,3,4,5,7],[6],[8]]
=> {{1,2,3,4,5,7},{6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,4,7,6,5,8] => [[1,2,3,4,5,8],[6],[7]]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,6,8,5] => [[1,2,3,4,5,8],[6],[7]]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,6,5] => [[1,2,3,4,5,8],[6],[7]]
=> {{1,2,3,4,5,8},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,7,6,5] => [[1,2,3,4,5],[6],[7],[8]]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,8,7,6] => [[1,2,3,4,6],[5,7],[8]]
=> {{1,2,3,4,6},{5,7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,3,5,6,8,7,4] => [[1,2,3,4,6,7],[5],[8]]
=> {{1,2,3,4,6,7},{5},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,3,5,7,6,4,8] => [[1,2,3,4,6,8],[5],[7]]
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,3,5,7,6,8,4] => [[1,2,3,4,6,8],[5],[7]]
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,3,5,7,8,6,4] => [[1,2,3,4,6,8],[5],[7]]
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3,5,8,7,6,4] => [[1,2,3,4,6],[5],[7],[8]]
=> {{1,2,3,4,6},{5},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,5,4,7,8] => [[1,2,3,4,7,8],[5],[6]]
=> {{1,2,3,4,7,8},{5},{6}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,3,6,5,4,8,7] => [[1,2,3,4,7],[5,8],[6]]
=> {{1,2,3,4,7},{5,8},{6}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,3,6,5,7,4,8] => [[1,2,3,4,7,8],[5],[6]]
=> {{1,2,3,4,7,8},{5},{6}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,5,7,8,4] => [[1,2,3,4,7,8],[5],[6]]
=> {{1,2,3,4,7,8},{5},{6}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,3,6,5,8,7,4] => [[1,2,3,4,7],[5,8],[6]]
=> {{1,2,3,4,7},{5,8},{6}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,6,7,5,4,8] => [[1,2,3,4,7,8],[5],[6]]
=> {{1,2,3,4,7,8},{5},{6}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,5,8,4] => [[1,2,3,4,7,8],[5],[6]]
=> {{1,2,3,4,7,8},{5},{6}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,6,7,8,5,4] => [[1,2,3,4,7,8],[5],[6]]
=> {{1,2,3,4,7,8},{5},{6}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,3,6,8,7,5,4] => [[1,2,3,4,7],[5],[6],[8]]
=> {{1,2,3,4,7},{5},{6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,7,6,5,4,8] => [[1,2,3,4,8],[5],[6],[7]]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,6,5,8,4] => [[1,2,3,4,8],[5],[6],[7]]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,3,7,6,8,5,4] => [[1,2,3,4,8],[5],[6],[7]]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,6,5,4] => [[1,2,3,4,8],[5],[6],[7]]
=> {{1,2,3,4,8},{5},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,3,5,8,7,6] => [[1,2,3,5,6],[4,7],[8]]
=> {{1,2,3,5,6},{4,7},{8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,8,7] => [[1,2,3,5,7],[4,6,8]]
=> {{1,2,3,5,7},{4,6,8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,2,4,3,6,8,7,5] => ?
=> ?
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,2,4,3,7,6,8,5] => [[1,2,3,5,8],[4,6],[7]]
=> {{1,2,3,5,8},{4,6},{7}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,8,7,6,5] => [[1,2,3,5],[4,6],[7],[8]]
=> {{1,2,3,5},{4,6},{7},{8}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,2,4,5,3,8,7,6] => [[1,2,3,5,6],[4,7],[8]]
=> {{1,2,3,5,6},{4,7},{8}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,6,8,7,3] => [[1,2,3,5,6,7],[4],[8]]
=> {{1,2,3,5,6,7},{4},{8}}
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,2,4,5,7,8,6,3] => [[1,2,3,5,6,8],[4],[7]]
=> {{1,2,3,5,6,8},{4},{7}}
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,2,4,5,8,7,6,3] => [[1,2,3,5,6],[4],[7],[8]]
=> {{1,2,3,5,6},{4},{7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,2,4,6,5,3,7,8] => [[1,2,3,5,7,8],[4],[6]]
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,2,4,6,7,5,8,3] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,2,4,6,7,8,5,3] => [[1,2,3,5,7,8],[4],[6]]
=> {{1,2,3,5,7,8},{4},{6}}
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,2,4,7,6,5,8,3] => [[1,2,3,5,8],[4],[6],[7]]
=> {{1,2,3,5,8},{4},{6},{7}}
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,2,4,7,6,8,5,3] => ?
=> ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,2,4,8,7,6,5,3] => [[1,2,3,5],[4],[6],[7],[8]]
=> {{1,2,3,5},{4},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,5,4,3,6,7,8] => [[1,2,3,6,7,8],[4],[5]]
=> {{1,2,3,6,7,8},{4},{5}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,5,4,3,6,8,7] => [[1,2,3,6,7],[4,8],[5]]
=> {{1,2,3,6,7},{4,8},{5}}
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,2,5,4,3,7,8,6] => [[1,2,3,6,8],[4,7],[5]]
=> {{1,2,3,6,8},{4,7},{5}}
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,2,5,4,6,7,3,8] => [[1,2,3,6,7,8],[4],[5]]
=> {{1,2,3,6,7,8},{4},{5}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,4,6,7,8,3] => [[1,2,3,6,7,8],[4],[5]]
=> {{1,2,3,6,7,8},{4},{5}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,2,5,4,6,8,7,3] => [[1,2,3,6,7],[4,8],[5]]
=> {{1,2,3,6,7},{4,8},{5}}
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,2,5,4,7,6,8,3] => [[1,2,3,6,8],[4,7],[5]]
=> {{1,2,3,6,8},{4,7},{5}}
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,2,5,4,7,8,6,3] => ?
=> ?
=> ? = 2
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000834
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [[1]]
=> [1] => 0
[1,0,1,0]
=> [1,2] => [[1,2]]
=> [1,2] => 1
[1,1,0,0]
=> [2,1] => [[1],[2]]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [[1,2,3]]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [[1,2],[3]]
=> [3,1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> [3,1,2] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> [2,4,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> [4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[1,3],[2],[4]]
=> [4,2,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [[1,3,4],[2,5]]
=> [2,5,1,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [[1,3,5],[2,4]]
=> [2,4,1,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[1,3,4],[2,5]]
=> [2,5,1,3,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> [5,2,4,1,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [[1,2,3,5],[4,6],[7]]
=> [7,4,6,1,2,3,5] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [[1,2,3,6],[4,7],[5]]
=> [5,4,7,1,2,3,6] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [[1,2,3,5],[4,6],[7]]
=> [7,4,6,1,2,3,5] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [[1,2,4,5],[3,6],[7]]
=> [7,3,6,1,2,4,5] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,6,4] => [[1,2,4,5],[3,6],[7]]
=> [7,3,6,1,2,4,5] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => [[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,5,7,4] => [[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,5,4] => [[1,2,4,5],[3,6],[7]]
=> [7,3,6,1,2,4,5] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,6,5,4] => [[1,2,4],[3,5],[6],[7]]
=> [7,6,3,5,1,2,4] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,6,5] => [[1,2,3,5],[4,6],[7]]
=> [7,4,6,1,2,3,5] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,5,2,7] => [[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,4,2,6,7] => [[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,4,2,7,6] => [[1,2,3,6],[4,7],[5]]
=> [5,4,7,1,2,3,6] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,4,6,2,7] => [[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,4,7,6,2] => [[1,2,3,5],[4,6],[7]]
=> [7,4,6,1,2,3,5] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,5,6,4,2,7] => [[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,5,4,2,7] => [[1,2,3,7],[4],[5],[6]]
=> [6,5,4,1,2,3,7] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [[1,2,3,6],[4],[5],[7]]
=> [7,5,4,1,2,3,6] => ? = 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,6,5,7,4,2] => [[1,2,3,5],[4],[6],[7]]
=> [7,6,4,1,2,3,5] => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7] => [[1,2,5,6,7],[3],[4]]
=> [4,3,1,2,5,6,7] => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,4,3,2,5,7,6] => [[1,2,5,6],[3,7],[4]]
=> [4,3,7,1,2,5,6] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,3,2,6,5,7] => [[1,2,5,7],[3,6],[4]]
=> [4,3,6,1,2,5,7] => ? = 2
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,4,3,2,6,7,5] => [[1,2,5,6],[3,7],[4]]
=> [4,3,7,1,2,5,6] => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,3,2,7,6,5] => [[1,2,5],[3,6],[4,7]]
=> [4,7,3,6,1,2,5] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,4,3,5,2,6,7] => [[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,3,5,2,7,6] => [[1,2,4,6],[3,7],[5]]
=> [5,3,7,1,2,4,6] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,4,3,5,6,2,7] => [[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => ? = 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,3,5,7,6,2] => [[1,2,4,5],[3,6],[7]]
=> [7,3,6,1,2,4,5] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,4,3,6,5,2,7] => [[1,2,4,7],[3,5],[6]]
=> [6,3,5,1,2,4,7] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,3,6,5,7,2] => [[1,2,4,6],[3,5],[7]]
=> [7,3,5,1,2,4,6] => ? = 2
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000845
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => ([],2)
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => ([(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => ([],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => ([(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => ([],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [5,7,1,2,3,4,6] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [6,7,5,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [4,7,1,2,3,5,6] => ([(0,5),(1,4),(1,6),(2,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [4,6,1,2,3,5,7] => ([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(6,5)],7)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [4,6,7,1,2,3,5] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [7,4,6,1,2,3,5] => ([(1,4),(2,3),(2,6),(4,5),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [4,5,7,1,2,3,6] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [6,4,5,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [5,7,4,6,1,2,3] => ([(0,6),(1,4),(2,3),(2,6),(4,5)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [5,6,4,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [5,6,7,4,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [7,5,6,4,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [6,7,5,4,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [3,7,1,2,4,5,6] => ([(0,4),(1,3),(1,6),(4,6),(5,2),(6,5)],7)
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [3,6,1,2,4,5,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,5),(6,2)],7)
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [3,6,7,1,2,4,5] => ([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [7,3,6,1,2,4,5] => ([(1,4),(2,3),(2,6),(4,6),(6,5)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [3,5,1,2,4,6,7] => ([(0,4),(1,3),(1,5),(3,6),(4,5),(5,6),(6,2)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [3,5,7,1,2,4,6] => ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [3,5,6,1,2,4,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,2),(6,5)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [3,5,6,7,1,2,4] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,6,4] => [7,3,5,6,1,2,4] => ([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => [6,3,5,1,2,4,7] => ([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(5,6)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,5,7,4] => [6,3,5,7,1,2,4] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,5,4] => [6,7,3,5,1,2,4] => ([(0,3),(1,5),(2,4),(2,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,6,5,4] => [7,6,3,5,1,2,4] => ([(2,5),(3,4),(3,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => [3,4,7,1,2,5,6] => ([(0,4),(1,5),(4,6),(5,2),(5,6),(6,3)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5,7] => [3,4,6,1,2,5,7] => ([(0,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [3,4,6,7,1,2,5] => ([(0,5),(1,3),(3,6),(4,2),(5,4),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,6,5] => [7,3,4,6,1,2,5] => ([(1,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [3,4,5,7,1,2,6] => ([(0,5),(1,3),(3,6),(4,2),(4,6),(5,4)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,6,2] => [7,3,4,5,6,1,2] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,5,7,2] => [6,3,4,5,7,1,2] => ([(0,6),(1,3),(2,4),(4,5),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,6,7,5,2] => [6,7,3,4,5,1,2] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [7,6,3,4,5,1,2] => ([(2,4),(3,5),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,4,2,7,6] => [5,3,4,7,1,2,6] => ([(0,5),(0,6),(1,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,4,6,7,2] => [5,3,4,6,7,1,2] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,4,7,6,2] => [5,7,3,4,6,1,2] => ([(0,3),(1,5),(2,4),(2,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,4,7,2] => [5,6,3,4,7,1,2] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,4,2] => [5,6,7,3,4,1,2] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 1
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000451
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 83%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [2,1] => 2 = 1 + 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,4,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,1,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,5,2] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [2,1,4,3,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,1,3,5,4] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,5,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,5,4,2] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,5,3,1] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,1,5,3,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,5,4] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [6,5,4,3,2,7,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1,5] => [4,3,2,1,7,6,5] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,1,2] => [6,5,4,3,1,7,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,6,1] => [5,4,3,2,7,6,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [6,5,4,3,7,2,1] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1,5,7] => [4,3,2,1,6,5,7] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,1,5,6] => [4,3,2,1,5,7,6] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,7,1] => [5,4,3,2,6,7,1] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,4] => [3,2,1,7,6,5,4] => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,1,3] => [6,5,4,2,1,7,3] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,1,2] => [5,4,3,1,7,6,2] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [6,5,4,2,7,3,1] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,1,2,7] => [5,4,3,1,6,2,7] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => [4,3,2,7,6,5,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,1,2] => [6,5,4,1,7,3,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,6,2,1] => [5,4,3,7,6,2,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [6,5,4,7,3,2,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,1,7] => [4,3,2,6,5,1,7] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,5,1,6] => [4,3,2,5,1,7,6] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [5,4,3,6,2,7,1] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,1,6,7] => [4,3,2,5,1,6,7] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,1,4,7] => [3,2,1,6,5,4,7] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,1,4,5] => [3,2,1,6,4,7,5] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,1,4,6] => [3,2,1,5,4,7,6] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1,4,5] => [3,2,1,4,7,6,5] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,1,2] => [5,4,3,6,1,7,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,7,1] => [4,3,2,6,5,7,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,5,6,1] => [4,3,2,5,7,6,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [5,4,3,6,7,2,1] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,1,7] => [4,3,2,5,6,1,7] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => [3,2,1,5,4,6,7] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,2,1,4,5,7] => [3,2,1,4,6,5,7] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => [3,2,1,4,5,7,6] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,1,4] => [6,5,3,2,1,7,4] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,1,3] => [5,4,2,1,7,6,3] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [6,5,3,2,7,4,1] => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,1,3,7] => [5,4,2,1,6,3,7] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,1,2] => [4,3,1,7,6,5,2] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,1,2] => [6,5,3,1,7,4,2] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,3,1] => [5,4,2,7,6,3,1] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [6,5,3,7,4,2,1] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,1,7] => [5,4,2,6,3,1,7] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,1,2,7] => [4,3,1,6,5,2,7] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,1,2,6] => [4,3,1,5,2,7,6] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,7,1] => [5,4,2,6,3,7,1] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,3,5,1,2,6,7] => [4,3,1,5,2,6,7] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,4,1] => [3,2,7,6,5,4,1] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,1,3] => [6,5,2,1,7,4,3] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,1,2] => [5,4,1,7,6,3,2] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [6,5,2,7,4,3,1] => ? = 2 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St000455
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1] => ([],1)
=> ? = 0 - 1
[1,0,1,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> -1 = 0 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 1 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> -1 = 0 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 1 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [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
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [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)
=> -1 = 0 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [6,5,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [4,1,2,3,5,6] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [4,6,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => [5,3,4,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => [4,3,5,1,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => [4,3,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,2] => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,3,6,2] => [4,5,3,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => [5,4,3,6,1,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [5,4,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => [6,2,5,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => [6,2,4,5,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => [5,2,4,1,3,6] => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => [5,2,4,6,1,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,4,3] => [5,6,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [6,5,2,4,1,3] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 2 - 1
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: St000028
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 83%
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [3,1,2] => [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,4,2,1] => [1,4,3,2] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [4,1,3,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,3,1] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,3,4,1] => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,3,4,2] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,5,3,2,1] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,3,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,5,4,3,2] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [1,2,5,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [5,4,1,3,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [2,5,1,4,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [5,1,4,3,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,5,4,3,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,5,3,1] => [1,2,5,4,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,3,4,1] => [1,5,2,4,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,3,5,4,1] => [1,2,5,4,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,5,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [2,5,4,1,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [5,2,4,1,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [2,5,4,1,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,4,5,1,2] => [1,3,5,2,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [2,5,1,4,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,2,5,4,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [5,1,3,4,2] => [5,1,2,4,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,5,2,4,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,2,5,4,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,2,3,5,4] => 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,2,7,6,5,4,3] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,1,6,5,4,3,2] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,2,7,6,5,4,3] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,1,2,6,5,4,3] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,7,2,6,5,4,3] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,2,7,6,5,4,3] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,2,3,7,6,5,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,2,6,1,5,4,3] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,3,7,2,6,5,4] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,7,1,6,5,4,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [7,1,6,5,4,3,2] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,2,7,6,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [7,1,2,6,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,7,2,6,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,2,7,6,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,2,3,7,6,5,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,1,2,5,4,3] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [3,7,1,2,6,5,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [7,1,6,2,5,4,3] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [1,7,6,2,5,4,3] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => [1,3,7,2,6,5,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => [7,1,2,6,5,4,3] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => [1,7,2,6,5,4,3] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [3,4,7,6,5,2,1] => [1,2,7,6,5,4,3] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [3,4,6,7,5,2,1] => [1,2,3,7,6,5,4] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => [7,1,2,3,6,5,4] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [3,7,4,5,6,2,1] => [1,7,2,3,6,5,4] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [3,4,7,5,6,2,1] => [1,2,7,3,6,5,4] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [3,4,5,7,6,2,1] => [1,2,3,7,6,5,4] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => [1,2,3,4,7,6,5] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => [7,6,5,4,1,3,2] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [6,7,5,4,2,3,1] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => [7,2,6,5,1,4,3] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [5,7,6,4,2,3,1] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [5,6,7,4,2,3,1] => [1,3,7,6,2,5,4] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [7,6,4,5,2,3,1] => [7,6,2,5,1,4,3] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => [3,7,2,6,1,5,4] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [7,4,6,5,2,3,1] => [7,2,6,5,1,4,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [4,7,6,5,2,3,1] => [2,7,6,5,1,4,3] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,6,4] => [4,6,7,5,2,3,1] => [1,3,7,6,2,5,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => [7,4,5,6,2,3,1] => [7,1,3,6,2,5,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,5,7,4] => [4,7,5,6,2,3,1] => [1,7,3,6,2,5,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,5,4] => [4,5,7,6,2,3,1] => [1,3,7,6,2,5,4] => ? = 2
Description
The number of stack-sorts needed to sort a permutation.
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Matching statistic: St001208
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[1,1,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,2] => 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,2,6,4] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,2,4,5] => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => ? = 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ? = 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,5] => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,2,3] => ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ? = 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,6,4] => ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => ? = 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,2,3,4] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => ? = 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => ? = 2
Description
The 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 $K[x]/(x^n)$.
Matching statistic: St001545
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> ? = 0 + 1
[1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0]
=> [1,2] => ([],2)
=> ([],1)
=> ? = 0 + 1
[1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ([],1)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$
els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k
$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000374The number of exclusive right-to-left minima of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000782The indicator function of whether a given perfect matching is an L & P matching. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001597The Frobenius rank of a skew partition. St001624The breadth of a lattice. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000628The balance of a binary word. 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.
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