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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000036
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000036: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,4,2] => 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,3,2] => 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,5,4,1,2] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,5,1,4,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,1,5,3,2] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,5,1,3,2] => 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => [3,6,5,4,1,2] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [3,6,5,1,4,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => [3,6,1,5,4,2] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => [3,6,5,1,4,2] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => [3,6,1,5,4,2] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => [3,1,6,5,4,2] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4,6] => [3,1,6,5,4,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [3,1,6,5,4,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,3,5] => [1,6,5,4,3,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [1,6,5,4,3,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => [4,1,6,5,3,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,3,6] => [4,1,6,5,3,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => [4,6,1,5,3,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => [4,6,5,1,3,2] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => [4,6,1,5,3,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,3,5] => [4,1,6,5,3,2] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,6,1,2,3,5] => [4,6,1,5,3,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [4,1,6,5,3,2] => 1
Description
The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation.
These are multiplicities of Verma modules.
Matching statistic: St001022
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[]
=> []
=> [] => []
=> ? = 1 - 1
Description
Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001113
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St001113: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 - 1
Description
Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.
Matching statistic: St001181
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001181: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 75%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001181: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [5,2,3,1,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,2,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,1,7] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,1,7] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,1,7] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,1,7] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,1,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,1,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,1,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,1,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,1,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,1,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,1,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,1,2,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,1,2,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,1,2,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,1,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,1,3,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,1,3,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,1,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [5,6,3,1,2,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [5,6,4,1,2,3,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [5,4,6,1,2,3,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [5,6,2,1,3,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,6,1,2,3,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,3,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,4,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,5,2,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,4,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
Description
Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001186
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001186: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 75%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001186: Dyck paths ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [5,2,3,1,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,3,2,1,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [5,3,1,2,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,2,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,1,7] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,1,7] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,1,7] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,1,7] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,1,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,1,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,1,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,2,1,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,1,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,1,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,1,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,1,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,1,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,1,2,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,1,2,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [5,3,4,6,1,2,7] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,1,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,1,3,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,1,3,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,1,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [5,6,3,1,2,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [5,6,4,1,2,3,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [5,4,6,1,2,3,7] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [5,6,2,1,3,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,6,1,2,3,4,7] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,3,5,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,4,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,5,2,3,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,4,6,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 - 1
Description
Number of simple modules with grade at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001878
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 31%●distinct values known / distinct values provided: 25%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 31%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1,0]
=> ([],1)
=> ([],1)
=> ? = 1
[1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> ([],1)
=> ? = 1
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ([],1)
=> ? = 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ([],1)
=> ? = 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ([],1)
=> ? = 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? = 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? = 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ([],1)
=> ? = 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> ([],1)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? = 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ? = 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> ([],1)
=> ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ? = 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ? = 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ([],1)
=> ? = 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? = 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ? = 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([],1)
=> ? = 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ? = 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ? = 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? = 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ? = 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? = 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [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),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ([(3,6),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(4,5)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,1),(4,5)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 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: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 31%●distinct values known / distinct values provided: 25%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 31%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1,0]
=> ([],1)
=> ([],1)
=> ? = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> ([],1)
=> ? = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 1 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ([],1)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ([],1)
=> ? = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ([],1)
=> ? = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [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),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ([(3,6),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(4,5)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,1),(4,5)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 31%●distinct values known / distinct values provided: 25%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 31%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1,0]
=> ([],1)
=> ([],1)
=> ? = 1 - 1
[1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> ([],1)
=> ? = 1 - 1
[1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ([],1)
=> ? = 1 - 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ([],1)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> ([],1)
=> ? = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ([],1)
=> ? = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(2,4),(3,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([],1)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1)],2)
=> ? = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([],1)
=> ? = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ([(2,5),(3,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [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),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ([(3,6),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(4,6),(5,1),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,4),(1,6),(2,3),(2,4),(3,6),(4,5),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(4,5)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(3,6),(4,1),(4,5)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(3,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,1),(4,5),(4,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7)
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001906
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 75%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 0 = 1 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,3,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,4,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,3,2] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,5,4,1,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,5,1,4,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,5,1,4,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,5,4,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,1,5,3,2] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,5,1,3,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,5,3,2] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,5,4,3,2] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => [3,6,5,4,1,2] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [3,6,5,1,4,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,5] => [3,6,1,5,4,2] => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => [3,6,5,1,4,2] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,1,2,5,6] => [3,6,1,5,4,2] => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [3,1,5,6,2,4] => [3,1,6,5,4,2] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4,6] => [3,1,6,5,4,2] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [3,5,1,6,2,4] => [3,6,1,5,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => [3,6,5,1,4,2] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4,6] => [3,6,1,5,4,2] => 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [3,1,2,6,4,5] => [3,1,6,5,4,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [3,6,1,2,4,5] => [3,6,1,5,4,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,1,2,4,5,6] => [3,1,6,5,4,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => [1,6,5,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => [1,6,5,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,4,2,6,3,5] => [1,6,5,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,4,6,2,3,5] => [1,6,5,4,3,2] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => [1,6,5,4,3,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => [4,1,6,5,3,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,3,6] => [4,1,6,5,3,2] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => [4,6,1,5,3,2] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => [4,6,5,1,3,2] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => [4,6,1,5,3,2] => 1 = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [4,1,6,2,3,5] => [4,1,6,5,3,2] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,6,1,2,3,5] => [4,6,1,5,3,2] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [4,1,2,3,5,6] => [4,1,6,5,3,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,2,3] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,4,5,6,2,3,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,4,5,2,7,3,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,4,5,7,2,3,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,4,5,2,3,6,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,4,2,6,7,3,5] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,2,6,3,5,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,4,6,2,7,3,5] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,4,6,7,2,3,5] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,4,2,3,7,5,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,4,2,7,3,5,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,4,7,2,3,5,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,2,3,5,6,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,2,5,6,7,3,4] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,2,5,6,3,4,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,2,5,3,7,4,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,2,5,7,3,4,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,6,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,5,2,6,7,3,4] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,5,2,6,3,4,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,4] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,2,3,4] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,5,6,2,3,4,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,5,2,3,7,4,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,5,2,7,3,4,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,5,7,2,3,4,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,3,4,6,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,2,3,6,7,4,5] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,2,3,6,4,5,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,2,6,3,7,4,5] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,2,6,7,3,4,5] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,2,6,3,4,5,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,6,2,3,7,4,5] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,6,2,7,3,4,5] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,6,7,2,3,4,5] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,4,1,6,7,8,2,5] => [3,8,1,7,6,5,4,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [3,4,1,6,7,2,5,8] => [3,8,1,7,6,5,4,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,4,1,6,2,8,5,7] => [3,8,1,7,6,5,4,2] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [3,4,1,6,8,2,5,7] => [3,8,1,7,6,5,4,2] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [3,4,1,2,7,8,5,6] => [3,8,1,7,6,5,4,2] => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,4,1,2,7,5,6,8] => [3,8,1,7,6,5,4,2] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,4,1,2,5,8,6,7] => [3,8,1,7,6,5,4,2] => ? = 2 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [3,4,1,2,8,5,6,7] => [3,8,1,7,6,5,4,2] => ? = 2 - 1
Description
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
Let $\pi$ be a permutation. Its total displacement [[St000830]] is $D(\pi) = \sum_i |\pi(i) - i|$, and its absolute length [[St000216]] is the minimal number $T(\pi)$ of transpositions whose product is $\pi$. Finally, let $I(\pi)$ be the number of inversions [[St000018]] of $\pi$.
This statistic equals $\left(D(\pi)-T(\pi)-I(\pi)\right)/2$.
Diaconis and Graham [1] proved that this statistic is always nonnegative.
Matching statistic: St001964
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 0 = 1 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [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)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [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)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [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)
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [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)
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [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)
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [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)
=> ? = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [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)
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [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)
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [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)
=> ? = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [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)
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [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)
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [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)
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [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)
=> ? = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [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
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [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)
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [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
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [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)
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [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)
=> ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [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
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [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)
=> ? = 2 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [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)
=> ? = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [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)
=> ? = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [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
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [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
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [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 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [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 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [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 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [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 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [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
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [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)
=> ? = 2 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [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
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [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
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [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
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,8),(4,3),(4,7),(5,2),(5,7),(6,4),(6,5),(7,8),(7,9),(8,10),(9,10),(10,1)],11)
=> ? = 1 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(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
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(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
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 - 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 1 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1 - 1
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 1 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 1 - 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 1 - 1
[]
=> [1,0]
=> [1,0]
=> ([],1)
=> 0 = 1 - 1
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.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1.
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