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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000791
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000791: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2,1] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,2] => [2] => [1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => [1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => [1,1,0,0,1,0]
=> 1
Description
The number of pairs of left tunnels, one strictly containing the other, of a Dyck path.
The statistic counting all pairs of distinct tunnels is the area of a Dyck path [[St000012]].
Matching statistic: St001232
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
=> [2,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [2,3,1] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 3 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 3 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,3,5,1,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,3,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,6,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,6,3,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,5,1,3,6,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,3,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,4,2,6,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,1,4,6,2,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [3,4,1,2,6,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,5,2,6,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,5,1,2,6,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,8,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,8,5,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,3,8,5,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,6,8,3,5,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,3,7,5,8,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,4,3,7,8,5,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,4,7,3,5,8,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,4,7,8,3,5,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,1,3,6,5,8,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [2,4,1,3,6,8,5,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [2,4,1,6,3,5,8,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,8,5,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [2,4,1,6,8,3,5,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,4,6,1,3,5,8,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [2,4,6,1,3,8,5,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [2,4,6,1,8,3,5,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [2,4,1,3,7,5,8,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [2,4,1,3,7,8,5,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,4,1,7,3,5,8,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [2,4,1,7,3,8,5,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,4,1,7,8,3,5,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,4,7,1,8,3,5,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [2,1,5,3,6,4,8,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [2,1,5,3,6,8,4,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [2,1,5,6,3,4,8,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [2,1,5,6,8,3,4,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,7,4,8,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001330
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [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)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 6 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 + 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3 + 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 + 2
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 1 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,3,4,7,6,1,5] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [7,3,4,6,1,2,5] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,3,6,5,1,7,4] => ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => ([(0,4),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => ([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [7,4,1,5,6,2,3] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,7,1,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [2,3,5,1,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,1,5,6,7,8,3] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ([(0,7),(1,7),(2,7),(3,4),(4,6),(5,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,4,1,5,6,8,3,7] => ([(0,7),(1,7),(2,5),(3,4),(4,7),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [2,3,4,7,1,5,8,6] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,5,7,3,8,6] => ([(0,6),(1,5),(2,7),(3,5),(3,7),(4,6),(4,7)],8)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => ([(0,6),(1,7),(2,7),(3,4),(3,5),(4,6),(5,7)],8)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,5,1,3,6,7,8,4] => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [2,3,4,8,1,5,6,7] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ([(0,6),(1,6),(2,7),(3,7),(4,5),(4,7),(5,6)],8)
=> 2 = 0 + 2
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,4,1,5,8,3,6,7] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(4,7),(6,7)],8)
=> 2 = 0 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001964
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 0
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 0
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St001625
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001625: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001625: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ?
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ?
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ?
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,10),(1,14),(3,13),(4,18),(5,17),(6,12),(7,8),(7,17),(8,3),(8,11),(9,6),(9,16),(10,5),(10,7),(11,13),(11,14),(12,18),(13,15),(14,9),(14,15),(15,16),(16,4),(16,12),(17,1),(17,11),(18,2)],19)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,10),(1,16),(3,12),(4,11),(5,13),(6,14),(7,6),(7,12),(8,4),(8,17),(9,5),(9,15),(10,3),(10,7),(11,16),(12,9),(12,14),(13,17),(14,15),(15,8),(15,13),(16,2),(17,1),(17,11)],18)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ?
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,10),(1,17),(3,16),(4,11),(5,14),(6,15),(7,6),(7,12),(8,5),(8,11),(9,7),(9,18),(10,4),(10,8),(11,9),(11,14),(12,15),(12,16),(13,17),(14,18),(15,13),(16,1),(16,13),(17,2),(18,3),(18,12)],19)
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,10),(1,17),(3,16),(4,11),(5,14),(6,15),(7,6),(7,12),(8,5),(8,11),(9,7),(9,18),(10,4),(10,8),(11,9),(11,14),(12,15),(12,16),(13,17),(14,18),(15,13),(16,1),(16,13),(17,2),(18,3),(18,12)],19)
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,10),(1,16),(3,12),(4,11),(5,13),(6,14),(7,6),(7,12),(8,4),(8,17),(9,5),(9,15),(10,3),(10,7),(11,16),(12,9),(12,14),(13,17),(14,15),(15,8),(15,13),(16,2),(17,1),(17,11)],18)
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,10),(1,14),(3,13),(4,18),(5,17),(6,12),(7,8),(7,17),(8,3),(8,11),(9,6),(9,16),(10,5),(10,7),(11,13),(11,14),(12,18),(13,15),(14,9),(14,15),(15,16),(16,4),(16,12),(17,1),(17,11),(18,2)],19)
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ?
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ?
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,9),(2,15),(3,14),(4,11),(5,13),(6,7),(6,14),(7,5),(7,10),(8,1),(9,3),(9,6),(10,13),(10,15),(11,8),(12,11),(13,12),(14,2),(14,10),(15,4),(15,12)],16)
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,9),(1,12),(2,13),(4,11),(5,10),(6,1),(6,11),(7,3),(8,5),(8,14),(9,4),(9,6),(10,13),(11,8),(11,12),(12,14),(13,7),(14,2),(14,10)],15)
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,9),(1,12),(2,13),(4,11),(5,10),(6,1),(6,11),(7,3),(8,5),(8,14),(9,4),(9,6),(10,13),(11,8),(11,12),(12,14),(13,7),(14,2),(14,10)],15)
=> ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,9),(2,15),(3,14),(4,11),(5,13),(6,7),(6,14),(7,5),(7,10),(8,1),(9,3),(9,6),(10,13),(10,15),(11,8),(12,11),(13,12),(14,2),(14,10),(15,4),(15,12)],16)
=> ? = 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,9),(1,11),(2,13),(4,12),(5,12),(6,10),(7,6),(7,13),(8,4),(8,5),(9,2),(9,7),(10,11),(11,8),(12,3),(13,1),(13,10)],14)
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,9),(1,11),(2,13),(4,12),(5,12),(6,10),(7,6),(7,13),(8,4),(8,5),(9,2),(9,7),(10,11),(11,8),(12,3),(13,1),(13,10)],14)
=> ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,9),(2,13),(3,12),(4,11),(5,11),(6,10),(7,6),(7,12),(8,3),(8,7),(9,4),(9,5),(10,13),(11,8),(12,2),(12,10),(13,1)],14)
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ?
=> ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,9),(2,13),(3,12),(4,11),(5,11),(6,10),(7,6),(7,12),(8,3),(8,7),(9,4),(9,5),(10,13),(11,8),(12,2),(12,10),(13,1)],14)
=> ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
=> ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
=> ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
=> ? = 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0
Description
The Möbius invariant of a lattice.
The '''Möbius invariant''' of a lattice $L$ is the value of the Möbius function applied to least and greatest element, that is $\mu(L)=\mu_L(\hat{0},\hat{1})$, where $\hat{0}$ is the least element of $L$ and $\hat{1}$ is the greatest element of $L$.
For the definition of the Möbius function, see [[St000914]].
Matching statistic: St001621
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ?
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ?
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,10),(1,14),(3,13),(4,18),(5,17),(6,12),(7,8),(7,17),(8,3),(8,11),(9,6),(9,16),(10,5),(10,7),(11,13),(11,14),(12,18),(13,15),(14,9),(14,15),(15,16),(16,4),(16,12),(17,1),(17,11),(18,2)],19)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,10),(1,16),(3,12),(4,11),(5,13),(6,14),(7,6),(7,12),(8,4),(8,17),(9,5),(9,15),(10,3),(10,7),(11,16),(12,9),(12,14),(13,17),(14,15),(15,8),(15,13),(16,2),(17,1),(17,11)],18)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,10),(1,17),(3,16),(4,11),(5,14),(6,15),(7,6),(7,12),(8,5),(8,11),(9,7),(9,18),(10,4),(10,8),(11,9),(11,14),(12,15),(12,16),(13,17),(14,18),(15,13),(16,1),(16,13),(17,2),(18,3),(18,12)],19)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,10),(1,17),(3,16),(4,11),(5,14),(6,15),(7,6),(7,12),(8,5),(8,11),(9,7),(9,18),(10,4),(10,8),(11,9),(11,14),(12,15),(12,16),(13,17),(14,18),(15,13),(16,1),(16,13),(17,2),(18,3),(18,12)],19)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,10),(1,16),(3,12),(4,11),(5,13),(6,14),(7,6),(7,12),(8,4),(8,17),(9,5),(9,15),(10,3),(10,7),(11,16),(12,9),(12,14),(13,17),(14,15),(15,8),(15,13),(16,2),(17,1),(17,11)],18)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,10),(1,14),(3,13),(4,18),(5,17),(6,12),(7,8),(7,17),(8,3),(8,11),(9,6),(9,16),(10,5),(10,7),(11,13),(11,14),(12,18),(13,15),(14,9),(14,15),(15,16),(16,4),(16,12),(17,1),(17,11),(18,2)],19)
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ?
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,9),(2,15),(3,14),(4,11),(5,13),(6,7),(6,14),(7,5),(7,10),(8,1),(9,3),(9,6),(10,13),(10,15),(11,8),(12,11),(13,12),(14,2),(14,10),(15,4),(15,12)],16)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,9),(1,12),(2,13),(4,11),(5,10),(6,1),(6,11),(7,3),(8,5),(8,14),(9,4),(9,6),(10,13),(11,8),(11,12),(12,14),(13,7),(14,2),(14,10)],15)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,9),(1,12),(2,13),(4,11),(5,10),(6,1),(6,11),(7,3),(8,5),(8,14),(9,4),(9,6),(10,13),(11,8),(11,12),(12,14),(13,7),(14,2),(14,10)],15)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,9),(2,15),(3,14),(4,11),(5,13),(6,7),(6,14),(7,5),(7,10),(8,1),(9,3),(9,6),(10,13),(10,15),(11,8),(12,11),(13,12),(14,2),(14,10),(15,4),(15,12)],16)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,9),(1,11),(2,13),(4,12),(5,12),(6,10),(7,6),(7,13),(8,4),(8,5),(9,2),(9,7),(10,11),(11,8),(12,3),(13,1),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,9),(1,11),(2,13),(4,12),(5,12),(6,10),(7,6),(7,13),(8,4),(8,5),(9,2),(9,7),(10,11),(11,8),(12,3),(13,1),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,9),(2,13),(3,12),(4,11),(5,11),(6,10),(7,6),(7,12),(8,3),(8,7),(9,4),(9,5),(10,13),(11,8),(12,2),(12,10),(13,1)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,9),(2,13),(3,12),(4,11),(5,11),(6,10),(7,6),(7,12),(8,3),(8,7),(9,4),(9,5),(10,13),(11,8),(12,2),(12,10),(13,1)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
=> ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,7),(2,9),(3,9),(4,8),(5,8),(6,2),(6,3),(7,4),(7,5),(8,6),(9,1)],10)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,8),(2,10),(4,9),(5,1),(5,10),(6,7),(7,2),(7,5),(8,9),(9,3),(10,4),(10,8)],11)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,7),(2,9),(3,10),(4,8),(5,4),(5,10),(6,1),(7,3),(7,5),(8,9),(9,6),(10,2),(10,8)],11)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ?
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ?
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ?
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,10),(1,14),(3,13),(4,18),(5,17),(6,12),(7,8),(7,17),(8,3),(8,11),(9,6),(9,16),(10,5),(10,7),(11,13),(11,14),(12,18),(13,15),(14,9),(14,15),(15,16),(16,4),(16,12),(17,1),(17,11),(18,2)],19)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,10),(1,16),(3,12),(4,11),(5,13),(6,14),(7,6),(7,12),(8,4),(8,17),(9,5),(9,15),(10,3),(10,7),(11,16),(12,9),(12,14),(13,17),(14,15),(15,8),(15,13),(16,2),(17,1),(17,11)],18)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ?
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,10),(1,17),(3,16),(4,11),(5,14),(6,15),(7,6),(7,12),(8,5),(8,11),(9,7),(9,18),(10,4),(10,8),(11,9),(11,14),(12,15),(12,16),(13,17),(14,18),(15,13),(16,1),(16,13),(17,2),(18,3),(18,12)],19)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(0,10),(1,17),(3,16),(4,11),(5,14),(6,15),(7,6),(7,12),(8,5),(8,11),(9,7),(9,18),(10,4),(10,8),(11,9),(11,14),(12,15),(12,16),(13,17),(14,18),(15,13),(16,1),(16,13),(17,2),(18,3),(18,12)],19)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(0,10),(1,16),(3,12),(4,11),(5,13),(6,14),(7,6),(7,12),(8,4),(8,17),(9,5),(9,15),(10,3),(10,7),(11,16),(12,9),(12,14),(13,17),(14,15),(15,8),(15,13),(16,2),(17,1),(17,11)],18)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(0,10),(1,14),(3,13),(4,18),(5,17),(6,12),(7,8),(7,17),(8,3),(8,11),(9,6),(9,16),(10,5),(10,7),(11,13),(11,14),(12,18),(13,15),(14,9),(14,15),(15,16),(16,4),(16,12),(17,1),(17,11),(18,2)],19)
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ?
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ?
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,9),(2,15),(3,14),(4,11),(5,13),(6,7),(6,14),(7,5),(7,10),(8,1),(9,3),(9,6),(10,13),(10,15),(11,8),(12,11),(13,12),(14,2),(14,10),(15,4),(15,12)],16)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,9),(1,12),(2,13),(4,11),(5,10),(6,1),(6,11),(7,3),(8,5),(8,14),(9,4),(9,6),(10,13),(11,8),(11,12),(12,14),(13,7),(14,2),(14,10)],15)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ([(0,9),(1,12),(2,13),(4,11),(5,10),(6,1),(6,11),(7,3),(8,5),(8,14),(9,4),(9,6),(10,13),(11,8),(11,12),(12,14),(13,7),(14,2),(14,10)],15)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,9),(2,15),(3,14),(4,11),(5,13),(6,7),(6,14),(7,5),(7,10),(8,1),(9,3),(9,6),(10,13),(10,15),(11,8),(12,11),(13,12),(14,2),(14,10),(15,4),(15,12)],16)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,9),(1,11),(2,13),(4,12),(5,12),(6,10),(7,6),(7,13),(8,4),(8,5),(9,2),(9,7),(10,11),(11,8),(12,3),(13,1),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ([(0,9),(1,11),(2,13),(4,12),(5,12),(6,10),(7,6),(7,13),(8,4),(8,5),(9,2),(9,7),(10,11),(11,8),(12,3),(13,1),(13,10)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,9),(2,13),(3,12),(4,11),(5,11),(6,10),(7,6),(7,12),(8,3),(8,7),(9,4),(9,5),(10,13),(11,8),(12,2),(12,10),(13,1)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ?
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,9),(2,13),(3,12),(4,11),(5,11),(6,10),(7,6),(7,12),(8,3),(8,7),(9,4),(9,5),(10,13),(11,8),(12,2),(12,10),(13,1)],14)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(0,8),(2,14),(3,12),(4,10),(5,11),(6,3),(6,11),(7,4),(7,13),(8,9),(9,5),(9,6),(10,14),(11,7),(11,12),(12,13),(13,2),(13,10),(14,1)],15)
=> ? = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,15),(3,14),(4,13),(5,12),(6,7),(6,13),(7,5),(7,10),(8,9),(9,4),(9,6),(10,12),(10,14),(11,15),(12,11),(13,3),(13,10),(14,1),(14,11),(15,2)],16)
=> ? = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000741
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> ([],2)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? = 0 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
=> ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 6 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(2,8),(3,4),(3,10),(4,9),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ([(2,8),(3,4),(3,10),(4,9),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ([(2,8),(3,4),(3,10),(4,9),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ([(4,7),(5,6),(6,7)],8)
=> ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(2,9),(3,8),(4,6),(5,7),(6,8),(7,9),(8,9)],10)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ([(2,9),(3,8),(4,6),(5,7),(6,8),(7,9),(8,9)],10)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ([(4,7),(5,6)],8)
=> ? = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 1 = 0 + 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St001550
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001550: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
[1,0,1,0]
=> [1]
=> [[1]]
=> [1] => 0
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 0
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 0
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => ? = 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => ? = 0
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ? = 0
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 0
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 0
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 0
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 0
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> [15,13,14,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,5,6,7,8,1,2,3,4] => ? = 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> [14,12,13,9,10,11,6,7,8,1,2,3,4,5] => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> [13,11,12,8,9,10,5,6,7,1,2,3,4] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> [12,10,11,7,8,9,4,5,6,1,2,3] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> [14,12,13,10,11,6,7,8,9,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,5,6,7,8,1,2,3,4] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> [13,11,12,9,10,6,7,8,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,5,6,7,1,2,3,4] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> [12,10,11,8,9,6,7,1,2,3,4,5] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,5,6,1,2,3,4] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,4,5,1,2,3] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> [14,13,10,11,12,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,5,6,7,8,1,2,3,4] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> [13,12,9,10,11,6,7,8,1,2,3,4,5] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> [12,11,8,9,10,5,6,7,1,2,3,4] => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> [13,12,10,11,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> [12,11,9,10,5,6,7,8,1,2,3,4] => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> [12,11,9,10,6,7,8,1,2,3,4,5] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [10,9,7,8,4,5,6,1,2,3] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [11,10,8,9,6,7,1,2,3,4,5] => ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> [10,9,7,8,5,6,1,2,3,4] => ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => ? = 0
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 0
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> [12,11,10,6,7,8,9,1,2,3,4,5] => ? = 0
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [11,10,9,5,6,7,8,1,2,3,4] => ? = 0
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [11,10,9,6,7,8,1,2,3,4,5] => ? = 0
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> [10,9,8,5,6,7,1,2,3,4] => ? = 0
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 0
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 0
Description
The number of inversions between exceedances where the greater exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
Matching statistic: St000264
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
[1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 0 + 4
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0 + 4
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0 + 4
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0 + 4
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 4
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 4 = 0 + 4
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 4
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 0 + 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3 + 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 4
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1 + 4
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 4 = 0 + 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 1 + 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
=> 4 = 0 + 4
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 0 + 4
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 4 = 0 + 4
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 0 + 4
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 0 + 4
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 4
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 6 + 4
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3 + 4
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 3 + 4
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 3 + 4
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 4
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 4
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 1 + 4
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1 + 4
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1 + 4
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1 + 4
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1 + 4
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(0,3),(0,7),(1,2),(1,6),(2,8),(3,9),(4,5),(4,8),(4,9),(5,6),(5,7),(6,8),(7,9)],10)
=> ? = 1 + 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 4
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 4
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ([(0,1),(0,2),(1,8),(2,8),(3,4),(3,6),(4,7),(5,6),(5,8),(6,7),(7,8)],9)
=> ? = 0 + 4
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 0 + 4
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 4 = 0 + 4
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
[1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 4 = 0 + 4
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000455The second largest eigenvalue of a graph if it is integral. St001846The number of elements which do not have a complement in the lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001868The number of alignments of type NE of a signed permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001845The number of join irreducibles minus the rank of a lattice.
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