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Your data matches 39 different statistics following compositions of up to 3 maps.
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Matching statistic: St001392
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00108: Permutations ācycle typeā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St001392: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00108: Permutations ācycle typeā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St001392: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> 0
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,1]
=> [1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [4,1]
=> [1]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,1]
=> [1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [4,1]
=> [1]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 0
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> 0
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St000487
Mp00119: Dyck paths āto 321-avoiding permutation (Krattenthaler)ā¶ Permutations
Mp00089: Permutations āInverse Kreweras complementā¶ Permutations
Mp00089: Permutations āInverse Kreweras complementā¶ Permutations
St000487: Permutations ā¶ ā¤Result quality: 9% āvalues known / values provided: 9%ādistinct values known / distinct values provided: 50%
Mp00089: Permutations āInverse Kreweras complementā¶ Permutations
Mp00089: Permutations āInverse Kreweras complementā¶ Permutations
St000487: Permutations ā¶ ā¤Result quality: 9% āvalues known / values provided: 9%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [1,2] => [2,1] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,2,1] => [2,1,3] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,3,2] => [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,4,3,1] => [1,3,2,4] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,2,4,1] => [2,1,3,4] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,3,1] => [2,3,1,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,4,2,1] => [3,1,2,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,3,4,2] => [4,2,3,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,3,2] => [4,3,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,4,3] => [2,4,3,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,4,2] => [4,1,3,2] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,2,3] => [3,4,1,2] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,5,4,1] => [1,2,4,3,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,4,3,5,1] => [1,3,2,4,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,5,3,4,1] => [1,3,4,2,5] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [2,4,5,3,1] => [1,4,2,3,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,2,4,5,1] => [2,1,3,4,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2,5,4,1] => [2,1,4,3,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,2,3,5,1] => [2,3,1,4,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [5,2,3,4,1] => [2,3,4,1,5] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,4,2,5,1] => [3,1,2,4,5] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [4,5,2,3,1] => [3,4,1,2,5] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [3,4,5,2,1] => [4,1,2,3,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,3,4,5,2] => [5,2,3,4,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,3,5,4,2] => [5,2,4,3,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,3,5,2] => [5,3,2,4,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,5,3,4,2] => [5,3,4,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [1,4,5,3,2] => [5,4,2,3,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,4,5,3] => [2,5,3,4,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,4,3] => [2,5,4,3,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,5,4] => [2,3,5,4,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,4,2,5,3] => [3,5,2,4,1] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [1,5,2,3,4] => [3,4,5,2,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,4,5,2] => [5,1,3,4,2] => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,1,5,4,2] => [5,1,4,3,2] => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [4,1,3,5,2] => [5,3,1,4,2] => 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [4,1,2,5,3] => [3,5,1,4,2] => 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [5,1,2,4,3] => [3,5,4,1,2] => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [4,1,5,2,3] => [4,5,1,3,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [3,4,1,5,2] => [5,1,2,4,3] => 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [4,5,1,3,2] => [5,4,1,2,3] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,3,4,6,5,1] => [1,2,3,5,4,6] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,3,5,4,6,1] => [1,2,4,3,5,6] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [2,3,6,4,5,1] => [1,2,4,5,3,6] => 1 = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [2,4,5,6,3,7,1] => [1,5,2,3,4,6,7] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [2,4,5,7,3,6,1] => [1,5,2,3,6,4,7] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => [1,5,2,6,3,4,7] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [2,5,6,7,3,4,1] => [1,5,6,2,3,4,7] => ? = 0 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => [1,6,2,3,4,5,7] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [3,2,4,5,6,7,1] => [2,1,3,4,5,6,7] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [3,2,4,5,7,6,1] => [2,1,3,4,6,5,7] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [3,2,4,6,5,7,1] => [2,1,3,5,4,6,7] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [3,2,4,7,5,6,1] => [2,1,3,5,6,4,7] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => [3,2,4,6,7,5,1] => [2,1,3,6,4,5,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [3,2,5,4,6,7,1] => [2,1,4,3,5,6,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [3,2,5,4,7,6,1] => [2,1,4,3,6,5,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [3,2,6,4,5,7,1] => [2,1,4,5,3,6,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [3,2,7,4,5,6,1] => [2,1,4,5,6,3,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,4,6] => [3,2,6,4,7,5,1] => [2,1,4,6,3,5,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,4,5,7] => [3,2,5,6,4,7,1] => [2,1,5,3,4,6,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,4,7,5] => [3,2,5,7,4,6,1] => [2,1,5,3,6,4,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,4,5] => [3,2,6,7,4,5,1] => [2,1,5,6,3,4,7] => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6] => [3,2,5,6,7,4,1] => [2,1,6,3,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7] => [4,2,3,5,6,7,1] => [2,3,1,4,5,6,7] => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => [4,2,3,5,7,6,1] => [2,3,1,4,6,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5,7] => [4,2,3,6,5,7,1] => [2,3,1,5,4,6,7] => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [4,2,3,7,5,6,1] => [2,3,1,5,6,4,7] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,5,6] => [4,2,3,6,7,5,1] => [2,3,1,6,4,5,7] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [5,2,3,4,6,7,1] => [2,3,4,1,5,6,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [5,2,3,4,7,6,1] => [2,3,4,1,6,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [6,2,3,4,5,7,1] => [2,3,4,5,1,6,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [7,2,3,4,5,6,1] => [2,3,4,5,6,1,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,2,6] => [6,2,3,4,7,5,1] => [2,3,4,6,1,5,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,2,5,7] => [5,2,3,6,4,7,1] => [2,3,5,1,4,6,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,2,7,5] => [5,2,3,7,4,6,1] => [2,3,5,1,6,4,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,6,7,2,5] => [6,2,3,7,4,5,1] => [2,3,5,6,1,4,7] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => [5,2,3,6,7,4,1] => [2,3,6,1,4,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,2,4,6,7] => [4,2,5,3,6,7,1] => [2,4,1,3,5,6,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,2,4,7,6] => [4,2,5,3,7,6,1] => [2,4,1,3,6,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,2,6,4,7] => [4,2,6,3,5,7,1] => [2,4,1,5,3,6,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,2,6,7,4] => [4,2,7,3,5,6,1] => [2,4,1,5,6,3,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,2,7,4,6] => [4,2,6,3,7,5,1] => [2,4,1,6,3,5,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,5,6,2,4,7] => [5,2,6,3,4,7,1] => [2,4,5,1,3,6,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,2,7,4] => [5,2,7,3,4,6,1] => [2,4,5,1,6,3,7] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,2,4] => [6,2,7,3,4,5,1] => [2,4,5,6,1,3,7] => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,5,7,2,4,6] => [5,2,6,3,7,4,1] => [2,4,6,1,3,5,7] => ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,2,4,5,7] => [4,2,5,6,3,7,1] => [2,5,1,3,4,6,7] => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,2,4,7,5] => [4,2,5,7,3,6,1] => [2,5,1,3,6,4,7] => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,6,2,7,4,5] => [4,2,6,7,3,5,1] => [2,5,1,6,3,4,7] => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,4,5] => [5,2,6,7,3,4,1] => [2,5,6,1,3,4,7] => ? = 0 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,2,4,5,6] => [4,2,5,6,7,3,1] => [2,6,1,3,4,5,7] => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,2,3,5,6,7] => [3,4,2,5,6,7,1] => [3,1,2,4,5,6,7] => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,4,2,3,5,7,6] => [3,4,2,5,7,6,1] => [3,1,2,4,6,5,7] => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,2,3,6,5,7] => [3,4,2,6,5,7,1] => [3,1,2,5,4,6,7] => ? = 0 + 1
Description
The length of the shortest cycle of a permutation.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
Mp00149: Permutations āLehmer code rotationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000455: Graphs ā¶ ā¤Result quality: 7% āvalues known / values provided: 7%ādistinct values known / distinct values provided: 17%
Mp00149: Permutations āLehmer code rotationā¶ Permutations
Mp00160: Permutations āgraph of inversionsā¶ Graphs
St000455: Graphs ā¶ ā¤Result quality: 7% āvalues known / values provided: 7%ādistinct values known / distinct values provided: 17%
Values
[1,0,1,0]
=> [2,1] => [1,2] => ([],2)
=> ? = 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,2,3] => ([],3)
=> ? = 0
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [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
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([],6)
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [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
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [1,6,5,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [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)
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [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
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [6,5,1,4,2,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [5,6,1,4,2,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [1,2,6,5,3,4] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => [6,1,5,4,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [1,6,5,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,5,3,2,6,1] => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,3,4,2,5,1] => [1,5,6,4,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => [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)
=> ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,5,1] => [1,6,4,5,2,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1] => [1,5,4,6,2,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => [6,4,3,5,1,2] => ([(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)
=> ? = 0
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [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)
=> ? = 0
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [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)
=> ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [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)
=> ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [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)
=> ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [6,1,2,3,5,4] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [6,5,1,2,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,1,2] => [5,6,1,2,4,3] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,1,2] => [6,1,5,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,1,2] => [1,6,5,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,1,2] => [6,5,4,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,1,2] => [5,6,4,1,3,2] => ([(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
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,1,2] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,1,2] => [6,4,5,1,3,2] => ([(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
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,1,2] => [5,4,6,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,1,3] => [6,1,2,5,4,3] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,1,3] => [1,6,2,5,4,3] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001236
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00233: Dyck paths āskew partitionā¶ Skew partitions
Mp00181: Skew partitions ārow lengthsā¶ Integer compositions
Mp00039: Integer compositions ācomplementā¶ Integer compositions
St001236: Integer compositions ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 50%
Mp00181: Skew partitions ārow lengthsā¶ Integer compositions
Mp00039: Integer compositions ācomplementā¶ Integer compositions
St001236: Integer compositions ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 50%
Values
[1,0,1,0]
=> [[1,1],[]]
=> [1,1] => [2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [[1,1,1],[]]
=> [1,1,1] => [3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [[2,1],[]]
=> [2,1] => [1,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1,2] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> [1,1,1,1] => [4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> [2,1,1] => [1,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1,2,1] => [2,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> [3,1] => [1,1,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> [2,2,1] => [1,2,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1,2] => [3,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [2,2] => [1,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [1,3] => [2,1,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1,2,2] => [2,2,1] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> [2,2,2] => [1,2,2,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1] => [5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> [2,1,1,1] => [1,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1,2,1,1] => [2,3] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> [3,1,1] => [1,1,3] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> [2,2,1,1] => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1,2,1] => [3,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [2,2,1] => [1,2,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [1,3,1] => [2,1,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> [4,1] => [1,1,1,2] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [2,3,1] => [1,2,1,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1,2,2,1] => [2,2,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> [3,2,1] => [1,1,2,2] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> [2,2,2,1] => [1,2,2,2] => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> [3,3,1] => [1,1,2,1,2] => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1,2] => [4,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [2,1,2] => [1,3,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [1,2,2] => [2,2,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [3,2] => [1,1,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [2,2,2] => [1,2,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [1,1,3] => [3,1,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2,3] => [1,2,1,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [1,4] => [2,1,1,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [1,2,3] => [2,2,1,1] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> [2,2,3] => [1,2,2,1,1] => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1,2,2] => [3,2,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [2,2,2] => [1,2,2,1] => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [1,3,2] => [2,1,2,1] => 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1,2,2,2] => [2,2,2,1] => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> [3,2,2] => [1,1,2,2,1] => ? = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> [3,3,2] => [1,1,2,1,2,1] => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [1,3,3] => [2,1,2,1,1] => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> [2,3,3] => [1,2,1,2,1,1] => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1] => [6] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> [2,1,1,1,1] => [1,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1,2,1,1,1] => [2,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1,1],[]]
=> [3,1,1,1] => [1,1,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> [2,2,1,1,1] => [1,2,4] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1,2,1,1] => [3,3] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [2,2,1,1] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1,1],[2]]
=> [1,3,1,1] => [2,1,3] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[4,1,1],[]]
=> [4,1,1] => [1,1,1,3] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1,1],[1]]
=> [2,3,1,1] => [1,2,1,3] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1,2,2,1,1] => [2,2,3] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1,1],[]]
=> [3,2,1,1] => [1,1,2,3] => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> [2,2,2,1,1] => [1,2,2,3] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> [3,3,1,1] => [1,1,2,1,3] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1,2,1] => [4,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [2,1,2,1] => [1,3,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [1,2,2,1] => [2,2,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> [3,2,1] => [1,1,2,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> [2,2,2,1] => [1,2,2,2] => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [[4,4,1],[2]]
=> [2,4,1] => [1,2,1,1,2] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> [1,2,3,1] => [2,2,1,2] => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> [3,3,1] => [1,1,2,1,2] => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3,1],[1,1]]
=> [2,2,3,1] => [1,2,2,1,2] => ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> [3,4,1] => [1,1,2,1,1,2] => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1]]
=> [1,1,2,2,1] => [3,2,2] => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> [2,2,2,1] => [1,2,2,2] => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> [1,3,2,1] => [2,1,2,2] => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> [4,2,1] => [1,1,1,2,2] => ? = 0 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> [2,3,2,1] => [1,2,1,2,2] => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,1],[1]]
=> [1,2,2,2,1] => [2,2,2,2] => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> [3,2,2,1] => [1,1,2,2,2] => ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> [2,2,2,2,1] => [1,2,2,2,2] => ? = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> [3,3,2,1] => [1,1,2,1,2,2] => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3,1],[2]]
=> [1,3,3,1] => [2,1,2,1,2] => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> [4,3,1] => [1,1,1,2,1,2] => ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> [2,3,3,1] => [1,2,1,2,1,2] => ? = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> [4,4,1] => [1,1,1,2,1,1,2] => ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> [3,3,3,1] => [1,1,2,1,2,1,2] => ? = 0 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1,1]]
=> [2,2,1,2] => [1,2,3,1] => ? = 0 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> [2,3,2] => [1,2,1,2,1] => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> [1,2,2,2] => [2,2,2,1] => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> [3,2,2] => [1,1,2,2,1] => ? = 1 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> [2,2,2,2] => [1,2,2,2,1] => ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> [3,3,2] => [1,1,2,1,2,1] => ? = 1 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> [2,2,3] => [1,2,2,1,1] => ? = 2 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> [1,2,4] => [2,2,1,1,1] => ? = 0 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> [2,2,4] => [1,2,2,1,1,1] => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> [1,1,2,3] => [3,2,1,1] => ? = 0 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> [2,2,3] => [1,2,2,1,1] => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> [1,3,3] => [2,1,2,1,1] => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [1,2,2,3] => [2,2,2,1,1] => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [[4,3,3],[1,1]]
=> [3,2,3] => [1,1,2,2,1,1] => ? = 2 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [[4,4,3],[1,1]]
=> [3,3,3] => [1,1,2,1,2,1,1] => ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [1,3,4] => [2,1,2,1,1,1] => ? = 0 + 1
Description
The dominant dimension of the corresponding Comp-Nakayama algebra.
Matching statistic: St000002
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000002: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000002: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => 0
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 0
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,6,2,1,5,4] => 0
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,8,2,1,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [3,8,2,1,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,8,2,7,6,1,5,4] => ? = 0
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,8,7,3,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [4,8,7,3,6,5,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [3,10,2,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [3,10,2,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [3,10,2,9,8,7,6,5,4,1] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [4,10,9,3,2,1,8,7,6,5] => ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [4,10,9,3,2,1,8,7,6,5] => ? = 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [4,10,9,3,2,8,7,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [4,10,9,3,8,7,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [4,10,9,3,8,7,2,6,5,1] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [4,10,9,3,8,7,6,5,2,1] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [5,10,9,8,4,3,2,1,7,6] => ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [5,10,9,8,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,7,9,6,10,8,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,7,9,6,11,8,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,4,3,7,10,6,11,12,9,8,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [2,1,4,3,8,9,11,7,6,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,4,3,8,10,11,7,12,9,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [2,1,5,6,4,3,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [3,4,2,1,6,5,8,7,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [3,4,2,1,6,5,8,7,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> [3,4,2,1,6,5,9,10,8,7,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
=> [3,4,2,1,6,5,9,11,8,12,10,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> [3,4,2,1,6,5,10,11,12,9,8,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> [3,4,2,1,7,8,6,5,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> [3,4,2,1,7,8,6,5,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
=> [3,4,2,1,7,9,6,10,8,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
=> [3,4,2,1,7,9,6,11,8,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> [3,4,2,1,7,10,6,11,12,9,8,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> [3,4,2,1,8,9,10,7,6,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> [3,4,2,1,8,9,11,7,6,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> [3,4,2,1,8,10,11,7,12,9,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [3,4,2,1,9,10,11,12,8,7,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> [3,5,2,6,4,1,8,7,10,9,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)]
=> [3,5,2,6,4,1,8,7,11,12,10,9] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)]
=> [3,5,2,6,4,1,9,10,8,7,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]
=> [3,5,2,6,4,1,9,11,8,12,10,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> [3,5,2,6,4,1,10,11,12,9,8,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> [3,5,2,7,4,8,6,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)]
=> [3,5,2,7,4,8,6,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> [3,5,2,7,4,9,6,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> [3,5,2,8,4,9,10,7,6,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]
=> [3,5,2,8,4,10,11,7,12,9,6,1] => [3,12,2,11,10,9,8,7,6,5,4,1] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
=> [3,6,2,7,8,5,4,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> [3,6,2,7,8,5,4,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> [3,6,2,7,9,5,4,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
Description
The number of occurrences of the pattern 123 in a permutation.
Matching statistic: St000404
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000404: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000404: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => 0
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 0
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,6,2,1,5,4] => 0
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,8,2,1,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [3,8,2,1,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,8,2,7,6,1,5,4] => ? = 0
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,8,7,3,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [4,8,7,3,6,5,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [3,10,2,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [3,10,2,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [3,10,2,9,8,7,6,5,4,1] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [4,10,9,3,2,1,8,7,6,5] => ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [4,10,9,3,2,1,8,7,6,5] => ? = 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [4,10,9,3,2,8,7,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [4,10,9,3,8,7,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [4,10,9,3,8,7,2,6,5,1] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [4,10,9,3,8,7,6,5,2,1] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [5,10,9,8,4,3,2,1,7,6] => ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [5,10,9,8,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,7,9,6,10,8,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,7,9,6,11,8,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,4,3,7,10,6,11,12,9,8,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [2,1,4,3,8,9,11,7,6,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,4,3,8,10,11,7,12,9,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [2,1,5,6,4,3,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [3,4,2,1,6,5,8,7,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [3,4,2,1,6,5,8,7,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> [3,4,2,1,6,5,9,10,8,7,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
=> [3,4,2,1,6,5,9,11,8,12,10,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> [3,4,2,1,6,5,10,11,12,9,8,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> [3,4,2,1,7,8,6,5,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> [3,4,2,1,7,8,6,5,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
=> [3,4,2,1,7,9,6,10,8,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
=> [3,4,2,1,7,9,6,11,8,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> [3,4,2,1,7,10,6,11,12,9,8,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> [3,4,2,1,8,9,10,7,6,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> [3,4,2,1,8,9,11,7,6,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> [3,4,2,1,8,10,11,7,12,9,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [3,4,2,1,9,10,11,12,8,7,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> [3,5,2,6,4,1,8,7,10,9,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)]
=> [3,5,2,6,4,1,8,7,11,12,10,9] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)]
=> [3,5,2,6,4,1,9,10,8,7,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]
=> [3,5,2,6,4,1,9,11,8,12,10,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> [3,5,2,6,4,1,10,11,12,9,8,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> [3,5,2,7,4,8,6,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)]
=> [3,5,2,7,4,8,6,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> [3,5,2,7,4,9,6,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> [3,5,2,8,4,9,10,7,6,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]
=> [3,5,2,8,4,10,11,7,12,9,6,1] => [3,12,2,11,10,9,8,7,6,5,4,1] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
=> [3,6,2,7,8,5,4,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> [3,6,2,7,8,5,4,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> [3,6,2,7,9,5,4,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
Description
The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation.
A permutation avoids these two pattern if and only if it is an ''input-restricted deques'', see [1].
Matching statistic: St000405
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000405: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000405: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => 0
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 0
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,6,2,1,5,4] => 0
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,8,2,1,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [3,8,2,1,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,8,2,7,6,1,5,4] => ? = 0
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,8,7,3,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [4,8,7,3,6,5,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [3,10,2,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [3,10,2,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [3,10,2,9,8,7,6,5,4,1] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [4,10,9,3,2,1,8,7,6,5] => ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [4,10,9,3,2,1,8,7,6,5] => ? = 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [4,10,9,3,2,8,7,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [4,10,9,3,8,7,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [4,10,9,3,8,7,2,6,5,1] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [4,10,9,3,8,7,6,5,2,1] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [5,10,9,8,4,3,2,1,7,6] => ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [5,10,9,8,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,7,9,6,10,8,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,7,9,6,11,8,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,4,3,7,10,6,11,12,9,8,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [2,1,4,3,8,9,11,7,6,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,4,3,8,10,11,7,12,9,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [2,1,5,6,4,3,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [3,4,2,1,6,5,8,7,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [3,4,2,1,6,5,8,7,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> [3,4,2,1,6,5,9,10,8,7,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
=> [3,4,2,1,6,5,9,11,8,12,10,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> [3,4,2,1,6,5,10,11,12,9,8,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> [3,4,2,1,7,8,6,5,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> [3,4,2,1,7,8,6,5,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
=> [3,4,2,1,7,9,6,10,8,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
=> [3,4,2,1,7,9,6,11,8,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> [3,4,2,1,7,10,6,11,12,9,8,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> [3,4,2,1,8,9,10,7,6,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> [3,4,2,1,8,9,11,7,6,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> [3,4,2,1,8,10,11,7,12,9,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [3,4,2,1,9,10,11,12,8,7,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> [3,5,2,6,4,1,8,7,10,9,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)]
=> [3,5,2,6,4,1,8,7,11,12,10,9] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)]
=> [3,5,2,6,4,1,9,10,8,7,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]
=> [3,5,2,6,4,1,9,11,8,12,10,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> [3,5,2,6,4,1,10,11,12,9,8,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> [3,5,2,7,4,8,6,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)]
=> [3,5,2,7,4,8,6,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> [3,5,2,7,4,9,6,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> [3,5,2,8,4,9,10,7,6,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]
=> [3,5,2,8,4,10,11,7,12,9,6,1] => [3,12,2,11,10,9,8,7,6,5,4,1] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
=> [3,6,2,7,8,5,4,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> [3,6,2,7,8,5,4,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> [3,6,2,7,9,5,4,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
Description
The number of occurrences of the pattern 1324 in a permutation.
There is no explicit formula known for the number of permutations avoiding this pattern (denoted by $S_n(1324)$), but it is shown in [1], improving bounds in [2] and [3] that
$$\lim_{n \rightarrow \infty} \sqrt[n]{S_n(1324)} \leq 13.73718.$$
Matching statistic: St000408
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000408: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000408: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => 0
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 0
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,6,2,1,5,4] => 0
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,8,2,1,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [3,8,2,1,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,8,2,7,6,1,5,4] => ? = 0
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,8,7,3,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [4,8,7,3,6,5,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [3,10,2,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [3,10,2,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [3,10,2,9,8,7,6,5,4,1] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [4,10,9,3,2,1,8,7,6,5] => ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [4,10,9,3,2,1,8,7,6,5] => ? = 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [4,10,9,3,2,8,7,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [4,10,9,3,8,7,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [4,10,9,3,8,7,2,6,5,1] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [4,10,9,3,8,7,6,5,2,1] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [5,10,9,8,4,3,2,1,7,6] => ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [5,10,9,8,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,7,9,6,10,8,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,7,9,6,11,8,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,4,3,7,10,6,11,12,9,8,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [2,1,4,3,8,9,11,7,6,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,4,3,8,10,11,7,12,9,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [2,1,5,6,4,3,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [3,4,2,1,6,5,8,7,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [3,4,2,1,6,5,8,7,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> [3,4,2,1,6,5,9,10,8,7,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
=> [3,4,2,1,6,5,9,11,8,12,10,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> [3,4,2,1,6,5,10,11,12,9,8,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> [3,4,2,1,7,8,6,5,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> [3,4,2,1,7,8,6,5,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
=> [3,4,2,1,7,9,6,10,8,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
=> [3,4,2,1,7,9,6,11,8,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> [3,4,2,1,7,10,6,11,12,9,8,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> [3,4,2,1,8,9,10,7,6,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> [3,4,2,1,8,9,11,7,6,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> [3,4,2,1,8,10,11,7,12,9,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [3,4,2,1,9,10,11,12,8,7,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> [3,5,2,6,4,1,8,7,10,9,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)]
=> [3,5,2,6,4,1,8,7,11,12,10,9] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)]
=> [3,5,2,6,4,1,9,10,8,7,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]
=> [3,5,2,6,4,1,9,11,8,12,10,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> [3,5,2,6,4,1,10,11,12,9,8,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> [3,5,2,7,4,8,6,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)]
=> [3,5,2,7,4,8,6,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> [3,5,2,7,4,9,6,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> [3,5,2,8,4,9,10,7,6,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]
=> [3,5,2,8,4,10,11,7,12,9,6,1] => [3,12,2,11,10,9,8,7,6,5,4,1] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
=> [3,6,2,7,8,5,4,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> [3,6,2,7,8,5,4,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> [3,6,2,7,9,5,4,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
Description
The number of occurrences of the pattern 4231 in a permutation.
It is a necessary condition that a permutation $\pi$ avoids this pattern for the Schubert variety associated to $\pi$ to be smooth [2].
Matching statistic: St000441
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000441: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000441: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => 0
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 0
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,6,2,1,5,4] => 0
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,8,2,1,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [3,8,2,1,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,8,2,7,6,1,5,4] => ? = 0
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,8,7,3,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [4,8,7,3,6,5,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [3,10,2,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [3,10,2,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [3,10,2,9,8,7,6,5,4,1] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [4,10,9,3,2,1,8,7,6,5] => ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [4,10,9,3,2,1,8,7,6,5] => ? = 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [4,10,9,3,2,8,7,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [4,10,9,3,8,7,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [4,10,9,3,8,7,2,6,5,1] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [4,10,9,3,8,7,6,5,2,1] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [5,10,9,8,4,3,2,1,7,6] => ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [5,10,9,8,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,7,9,6,10,8,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,7,9,6,11,8,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,4,3,7,10,6,11,12,9,8,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [2,1,4,3,8,9,11,7,6,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,4,3,8,10,11,7,12,9,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [2,1,5,6,4,3,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [3,4,2,1,6,5,8,7,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [3,4,2,1,6,5,8,7,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> [3,4,2,1,6,5,9,10,8,7,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
=> [3,4,2,1,6,5,9,11,8,12,10,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> [3,4,2,1,6,5,10,11,12,9,8,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> [3,4,2,1,7,8,6,5,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> [3,4,2,1,7,8,6,5,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
=> [3,4,2,1,7,9,6,10,8,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
=> [3,4,2,1,7,9,6,11,8,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> [3,4,2,1,7,10,6,11,12,9,8,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> [3,4,2,1,8,9,10,7,6,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> [3,4,2,1,8,9,11,7,6,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> [3,4,2,1,8,10,11,7,12,9,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [3,4,2,1,9,10,11,12,8,7,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> [3,5,2,6,4,1,8,7,10,9,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)]
=> [3,5,2,6,4,1,8,7,11,12,10,9] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)]
=> [3,5,2,6,4,1,9,10,8,7,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]
=> [3,5,2,6,4,1,9,11,8,12,10,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> [3,5,2,6,4,1,10,11,12,9,8,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> [3,5,2,7,4,8,6,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)]
=> [3,5,2,7,4,8,6,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> [3,5,2,7,4,9,6,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> [3,5,2,8,4,9,10,7,6,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]
=> [3,5,2,8,4,10,11,7,12,9,6,1] => [3,12,2,11,10,9,8,7,6,5,4,1] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
=> [3,6,2,7,8,5,4,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> [3,6,2,7,8,5,4,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> [3,6,2,7,9,5,4,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000534
Mp00146: Dyck paths āto tunnel matchingā¶ Perfect matchings
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000534: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Mp00283: Perfect matchings ānon-nesting-exceedence permutationā¶ Permutations
Mp00068: Permutations āSimion-Schmidt mapā¶ Permutations
St000534: Permutations ā¶ ā¤Result quality: 3% āvalues known / values provided: 3%ādistinct values known / distinct values provided: 17%
Values
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 0
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,6,5,4,3] => 0
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,4,3] => 0
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [3,6,2,1,5,4] => 0
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [3,8,2,1,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [3,8,2,1,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [3,8,2,7,6,1,5,4] => ? = 0
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [4,8,7,3,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [4,8,7,3,6,5,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,7,8,6,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,7,9,6,10,8,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,5,6,4,3,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,5,6,4,3,9,10,8,7] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,5,7,4,8,6,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [3,4,2,1,6,5,8,7,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [3,4,2,1,6,5,9,10,8,7] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [3,4,2,1,7,8,6,5,10,9] => [3,10,2,1,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => [3,10,2,1,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [3,10,2,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => [3,10,2,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => [3,10,2,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [3,6,2,8,9,5,10,7,4,1] => [3,10,2,9,8,7,6,5,4,1] => ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [4,10,9,3,2,1,8,7,6,5] => ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => [4,10,9,3,2,1,8,7,6,5] => ? = 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [4,10,9,3,2,8,7,1,6,5] => ? = 0
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => [4,10,9,3,8,7,2,1,6,5] => ? = 0
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [4,6,7,3,9,5,2,10,8,1] => [4,10,9,3,8,7,2,6,5,1] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [4,7,8,3,9,10,6,5,2,1] => [4,10,9,3,8,7,6,5,2,1] => ? = 1
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => [5,10,9,8,4,3,2,1,7,6] => ? = 0
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [5,6,8,9,4,3,10,7,2,1] => [5,10,9,8,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,7,9,6,10,8,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
=> [2,1,4,3,7,9,6,11,8,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
=> [2,1,4,3,7,10,6,11,12,9,8,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> [2,1,4,3,8,9,11,7,6,12,10,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,4,3,8,10,11,7,12,9,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
=> [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
=> [2,1,5,6,4,3,9,11,8,12,10,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
=> [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
=> [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
=> [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
=> [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
=> [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
=> [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 0
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
=> [3,4,2,1,6,5,8,7,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
=> [3,4,2,1,6,5,8,7,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> [3,4,2,1,6,5,9,10,8,7,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
=> [3,4,2,1,6,5,9,11,8,12,10,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> [3,4,2,1,6,5,10,11,12,9,8,7] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> [3,4,2,1,7,8,6,5,10,9,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> [3,4,2,1,7,8,6,5,11,12,10,9] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9),(11,12)]
=> [3,4,2,1,7,9,6,10,8,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,9),(10,11)]
=> [3,4,2,1,7,9,6,11,8,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> [3,4,2,1,7,10,6,11,12,9,8,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> [3,4,2,1,8,9,10,7,6,5,12,11] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 0
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> [3,4,2,1,8,9,11,7,6,12,10,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,8),(9,10)]
=> [3,4,2,1,8,10,11,7,12,9,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> [3,4,2,1,9,10,11,12,8,7,6,5] => [3,12,2,1,11,10,9,8,7,6,5,4] => ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> [3,5,2,6,4,1,8,7,10,9,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)]
=> [3,5,2,6,4,1,8,7,11,12,10,9] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)]
=> [3,5,2,6,4,1,9,10,8,7,12,11] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 0
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]
=> [3,5,2,6,4,1,9,11,8,12,10,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> [3,5,2,6,4,1,10,11,12,9,8,7] => [3,12,2,11,10,1,9,8,7,6,5,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12)]
=> [3,5,2,7,4,8,6,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11)]
=> [3,5,2,7,4,8,6,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12)]
=> [3,5,2,7,4,9,6,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> [3,5,2,8,4,9,10,7,6,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10)]
=> [3,5,2,8,4,10,11,7,12,9,6,1] => [3,12,2,11,10,9,8,7,6,5,4,1] => ? = 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
=> [3,6,2,7,8,5,4,1,10,9,12,11] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> [3,6,2,7,8,5,4,1,11,12,10,9] => [3,12,2,11,10,9,8,1,7,6,5,4] => ? = 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9),(11,12)]
=> [3,6,2,7,9,5,4,10,8,1,12,11] => [3,12,2,11,10,9,8,7,6,1,5,4] => ? = 0
Description
The number of 2-rises of a permutation.
A 2-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+2 = \pi(i+1)$.
For 1-rises, or successions, see [[St000441]].
The following 29 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000546The number of global descents of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000035The number of left outer peaks of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000842The breadth of a permutation. St000862The number of parts of the shifted shape of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St001964The interval resolution global dimension of a poset. St000221The number of strong fixed points of a permutation. St000461The rix statistic of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000056The decomposition (or block) number of a permutation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001481The minimal height of a peak of a Dyck path. St000307The number of rowmotion orbits of a poset. St001549The number of restricted non-inversions between exceedances. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone.
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