Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St001218
(load all 4 compositions to match this statistic)
Mapping
St001218: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 3
[1,0,1,0]
 => 4
[1,1,0,0]
 => 4
[1,0,1,0,1,0]
 => 5
[1,0,1,1,0,0]
 => 5
[1,1,0,0,1,0]
 => 5
[1,1,0,1,0,0]
 => 6
[1,1,1,0,0,0]
 => 5
[1,0,1,0,1,0,1,0]
 => 6
[1,0,1,0,1,1,0,0]
 => 6
[1,0,1,1,0,0,1,0]
 => 6
[1,0,1,1,0,1,0,0]
 => 8
[1,0,1,1,1,0,0,0]
 => 6
[1,1,0,0,1,0,1,0]
 => 6
[1,1,0,0,1,1,0,0]
 => 6
[1,1,0,1,0,0,1,0]
 => 8
[1,1,0,1,0,1,0,0]
 => 8
[1,1,0,1,1,0,0,0]
 => 8
[1,1,1,0,0,0,1,0]
 => 6
[1,1,1,0,0,1,0,0]
 => 8
[1,1,1,0,1,0,0,0]
 => 8
[1,1,1,1,0,0,0,0]
 => 6
[1,0,1,0,1,0,1,0,1,0]
 => 7
[1,0,1,0,1,0,1,1,0,0]
 => 7
[1,0,1,0,1,1,0,0,1,0]
 => 7
[1,0,1,0,1,1,0,1,0,0]
 => 10
[1,0,1,0,1,1,1,0,0,0]
 => 7
[1,0,1,1,0,0,1,0,1,0]
 => 7
[1,0,1,1,0,0,1,1,0,0]
 => 7
[1,0,1,1,0,1,0,0,1,0]
 => 12
[1,0,1,1,0,1,0,1,0,0]
 => 10
[1,0,1,1,0,1,1,0,0,0]
 => 12
[1,0,1,1,1,0,0,0,1,0]
 => 7
[1,0,1,1,1,0,0,1,0,0]
 => 10
[1,0,1,1,1,0,1,0,0,0]
 => 12
[1,0,1,1,1,1,0,0,0,0]
 => 7
[1,1,0,0,1,0,1,0,1,0]
 => 7
[1,1,0,0,1,0,1,1,0,0]
 => 7
[1,1,0,0,1,1,0,0,1,0]
 => 7
[1,1,0,0,1,1,0,1,0,0]
 => 10
[1,1,0,0,1,1,1,0,0,0]
 => 7
[1,1,0,1,0,0,1,0,1,0]
 => 10
[1,1,0,1,0,0,1,1,0,0]
 => 10
[1,1,0,1,0,1,0,0,1,0]
 => 10
[1,1,0,1,0,1,0,1,0,0]
 => 12
[1,1,0,1,0,1,1,0,0,0]
 => 10
[1,1,0,1,1,0,0,0,1,0]
 => 10
[1,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => 12
[1,1,0,1,1,1,0,0,0,0]
 => 10
Description
Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. It returns zero in case there is no such k.
Matching statistic: St000031
Mapping
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 60%
Values
[1,0]
 => [(1,2)]
 => [2,1] => [2,1] => 1 = 3 - 2
[1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => [2,1,4,3] => 2 = 4 - 2
[1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => [4,3,2,1] => 2 = 4 - 2
[1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3 = 5 - 2
[1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,5,4,3] => 3 = 5 - 2
[1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [4,3,2,1,6,5] => 3 = 5 - 2
[1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,5,3,4,2,1] => 4 = 6 - 2
[1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [6,5,4,3,2,1] => 3 = 5 - 2
[1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 4 = 6 - 2
[1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,6,5] => 4 = 6 - 2
[1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => [2,1,6,5,4,3,8,7] => 4 = 6 - 2
[1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => [2,1,8,7,5,6,4,3] => ? = 8 - 2
[1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => [2,1,8,7,6,5,4,3] => 4 = 6 - 2
[1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => [4,3,2,1,6,5,8,7] => 4 = 6 - 2
[1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => [4,3,2,1,8,7,6,5] => 4 = 6 - 2
[1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => [6,5,3,4,2,1,8,7] => ? = 8 - 2
[1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => [8,5,3,7,2,6,4,1] => ? = 8 - 2
[1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => [3,6,2,7,8,5,4,1] => [8,7,3,6,5,4,2,1] => ? = 8 - 2
[1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => 4 = 6 - 2
[1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => [8,5,7,4,2,6,3,1] => ? = 8 - 2
[1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => [8,7,6,4,5,3,2,1] => ? = 8 - 2
[1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => [8,7,6,5,4,3,2,1] => 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 5 = 7 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => [2,1,4,3,6,5,10,9,8,7] => 5 = 7 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,7,8,6,5,10,9] => [2,1,4,3,8,7,6,5,10,9] => 5 = 7 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => [2,1,4,3,10,9,7,8,6,5] => ? = 10 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => [2,1,4,3,10,9,8,7,6,5] => 5 = 7 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,5,6,4,3,8,7,10,9] => [2,1,6,5,4,3,8,7,10,9] => 5 = 7 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,5,6,4,3,9,10,8,7] => [2,1,6,5,4,3,10,9,8,7] => 5 = 7 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10)]
 => [2,1,5,7,4,8,6,3,10,9] => [2,1,8,7,5,6,4,3,10,9] => ? = 12 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => [2,1,10,7,5,9,4,8,6,3] => ? = 10 - 2
[1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,5,8,4,9,10,7,6,3] => [2,1,10,9,5,8,7,6,4,3] => ? = 12 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [2,1,6,7,8,5,4,3,10,9] => [2,1,8,7,6,5,4,3,10,9] => 5 = 7 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => [2,1,10,7,9,6,4,8,5,3] => ? = 10 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => [2,1,10,9,8,6,7,5,4,3] => ? = 12 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => 5 = 7 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [3,4,2,1,6,5,8,7,10,9] => [4,3,2,1,6,5,8,7,10,9] => 5 = 7 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [3,4,2,1,6,5,9,10,8,7] => [4,3,2,1,6,5,10,9,8,7] => 5 = 7 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [3,4,2,1,7,8,6,5,10,9] => [4,3,2,1,8,7,6,5,10,9] => 5 = 7 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => [3,4,2,1,7,9,6,10,8,5] => [4,3,2,1,10,9,7,8,6,5] => ? = 10 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [3,4,2,1,8,9,10,7,6,5] => [4,3,2,1,10,9,8,7,6,5] => 5 = 7 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => [3,5,2,6,4,1,8,7,10,9] => [6,5,3,4,2,1,8,7,10,9] => ? = 10 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => [3,5,2,6,4,1,9,10,8,7] => [6,5,3,4,2,1,10,9,8,7] => ? = 10 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [3,5,2,7,4,8,6,1,10,9] => [8,5,3,7,2,6,4,1,10,9] => ? = 10 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => [10,5,3,7,2,9,4,8,6,1] => ? = 12 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => [3,5,2,8,4,9,10,7,6,1] => [10,5,3,9,2,8,7,6,4,1] => ? = 10 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [3,6,2,7,8,5,4,1,10,9] => [8,7,3,6,5,4,2,1,10,9] => ? = 10 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => [3,6,2,7,9,5,4,10,8,1] => [10,7,3,6,9,4,2,8,5,1] => ? = 0 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => [3,6,2,8,9,5,10,7,4,1] => [10,9,3,8,6,5,7,4,2,1] => ? = 12 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => [3,7,2,8,9,10,6,5,4,1] => [10,9,3,8,7,6,5,4,2,1] => ? = 10 - 2
[1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [4,5,6,3,2,1,8,7,10,9] => [6,5,4,3,2,1,8,7,10,9] => 5 = 7 - 2
[1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [4,5,6,3,2,1,9,10,8,7] => [6,5,4,3,2,1,10,9,8,7] => 5 = 7 - 2
[1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [4,5,7,3,2,8,6,1,10,9] => [8,5,7,4,2,6,3,1,10,9] => ? = 12 - 2
[1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => [10,5,7,4,2,9,3,8,6,1] => ? = 10 - 2
[1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => [4,5,8,3,2,9,10,7,6,1] => [10,5,9,4,2,8,7,6,3,1] => ? = 12 - 2
[1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [4,6,7,3,8,5,2,1,10,9] => [8,7,6,4,5,3,2,1,10,9] => ? = 12 - 2
[1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => [10,7,6,4,9,3,2,8,5,1] => ? = 12 - 2
[1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => [4,6,8,3,9,5,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 12 - 2
[1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => [4,7,8,3,9,10,6,5,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 12 - 2
[1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [5,6,7,8,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => 5 = 7 - 2
[1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => [10,7,6,9,5,3,2,8,4,1] => ? = 10 - 2
[1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => [5,6,8,9,4,3,10,7,2,1] => [10,9,8,6,5,4,7,3,2,1] => ? = 12 - 2
[1,1,1,1,0,1,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7)]
 => [5,7,8,9,4,10,6,3,2,1] => [10,9,8,7,5,6,4,3,2,1] => ? = 10 - 2
[1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [6,7,8,9,10,5,4,3,2,1] => [10,9,8,7,6,5,4,3,2,1] => 5 = 7 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 6 = 8 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => [2,1,4,3,6,5,8,7,12,11,10,9] => 6 = 8 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
 => [2,1,4,3,6,5,9,10,8,7,12,11] => [2,1,4,3,6,5,10,9,8,7,12,11] => 6 = 8 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => [2,1,4,3,6,5,12,11,9,10,8,7] => ? = 12 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => [2,1,4,3,6,5,12,11,10,9,8,7] => 6 = 8 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
 => [2,1,4,3,7,8,6,5,10,9,12,11] => [2,1,4,3,8,7,6,5,10,9,12,11] => 6 = 8 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
 => [2,1,4,3,7,8,6,5,11,12,10,9] => [2,1,4,3,8,7,6,5,12,11,10,9] => 6 = 8 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
 => [2,1,4,3,7,9,6,10,8,5,12,11] => [2,1,4,3,10,9,7,8,6,5,12,11] => ? = 18 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => [2,1,4,3,12,9,7,11,6,10,8,5] => ? = 12 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,11),(9,10)]
 => [2,1,4,3,7,10,6,11,12,9,8,5] => [2,1,4,3,12,11,7,10,9,8,6,5] => ? = 18 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
 => [2,1,4,3,8,9,10,7,6,5,12,11] => [2,1,4,3,10,9,8,7,6,5,12,11] => 6 = 8 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => [2,1,4,3,12,9,11,8,6,10,7,5] => ? = 12 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,4,3,8,10,11,7,12,9,6,5] => [2,1,4,3,12,11,10,8,9,7,6,5] => ? = 18 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,4,3,9,10,11,12,8,7,6,5] => [2,1,4,3,12,11,10,9,8,7,6,5] => 6 = 8 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
 => [2,1,5,6,4,3,8,7,10,9,12,11] => [2,1,6,5,4,3,8,7,10,9,12,11] => 6 = 8 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
 => [2,1,5,6,4,3,8,7,11,12,10,9] => [2,1,6,5,4,3,8,7,12,11,10,9] => 6 = 8 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
 => [2,1,5,6,4,3,9,10,8,7,12,11] => [2,1,6,5,4,3,10,9,8,7,12,11] => 6 = 8 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [(1,2),(3,6),(4,5),(7,12),(8,9),(10,11)]
 => [2,1,5,6,4,3,9,11,8,12,10,7] => [2,1,6,5,4,3,12,11,9,10,8,7] => ? = 12 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
 => [2,1,5,6,4,3,10,11,12,9,8,7] => [2,1,6,5,4,3,12,11,10,9,8,7] => 6 = 8 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,10),(11,12)]
 => [2,1,5,7,4,8,6,3,10,9,12,11] => [2,1,8,7,5,6,4,3,10,9,12,11] => ? = 18 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7),(9,12),(10,11)]
 => [2,1,5,7,4,8,6,3,11,12,10,9] => [2,1,8,7,5,6,4,3,12,11,10,9] => ? = 18 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)]
 => [2,1,5,7,4,9,6,10,8,3,12,11] => [2,1,10,7,5,9,4,8,6,3,12,11] => ? = 12 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => [2,1,12,7,5,9,4,11,6,10,8,3] => ? = 18 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,11),(9,10)]
 => [2,1,5,7,4,10,6,11,12,9,8,3] => [2,1,12,7,5,11,4,10,9,8,6,3] => ? = 12 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
 => [2,1,5,8,4,9,10,7,6,3,12,11] => [2,1,10,9,5,8,7,6,4,3,12,11] => ? = 18 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
 => [2,1,5,8,4,9,11,7,6,12,10,3] => [2,1,12,9,5,8,11,6,4,10,7,3] => ? = 0 - 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,8),(9,10)]
 => [2,1,5,8,4,10,11,7,12,9,6,3] => [2,1,12,11,5,10,8,7,9,6,4,3] => ? = 18 - 2
[1,0,1,1,0,1,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9)]
 => [2,1,5,9,4,10,11,12,8,7,6,3] => [2,1,12,11,5,10,9,8,7,6,4,3] => ? = 18 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
 => [2,1,6,7,8,5,4,3,10,9,12,11] => [2,1,8,7,6,5,4,3,10,9,12,11] => 6 = 8 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
 => [2,1,6,7,8,5,4,3,11,12,10,9] => [2,1,8,7,6,5,4,3,12,11,10,9] => 6 = 8 - 2
[1,0,1,1,1,0,0,1,0,0,1,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
 => [2,1,6,7,9,5,4,10,8,3,12,11] => [2,1,10,7,9,6,4,8,5,3,12,11] => ? = 18 - 2
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => [2,1,12,7,9,6,4,11,5,10,8,3] => ? = 12 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
 => [2,1,7,8,9,10,6,5,4,3,12,11] => [2,1,10,9,8,7,6,5,4,3,12,11] => 6 = 8 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [2,1,8,9,10,11,12,7,6,5,4,3] => [2,1,12,11,10,9,8,7,6,5,4,3] => 6 = 8 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)]
 => [3,4,2,1,6,5,8,7,10,9,12,11] => [4,3,2,1,6,5,8,7,10,9,12,11] => 6 = 8 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,12),(10,11)]
 => [3,4,2,1,6,5,8,7,11,12,10,9] => [4,3,2,1,6,5,8,7,12,11,10,9] => 6 = 8 - 2
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St001880
(load all 8 compositions to match this statistic)
Mapping
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1,0]
 => [.,.]
 => [.,.]
 => ([],1)
 => ? = 3 - 2
[1,0,1,0]
 => [.,[.,.]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 4 - 2
[1,1,0,0]
 => [[.,.],.]
 => [[.,.],.]
 => ([(0,1)],2)
 => ? = 4 - 2
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 3 = 5 - 2
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 3 = 5 - 2
[1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 3 = 5 - 2
[1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 6 - 2
[1,1,1,0,0,0]
 => [[[.,.],.],.]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3 = 5 - 2
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 6 - 2
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 6 - 2
[1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 6 - 2
[1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 8 - 2
[1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 6 - 2
[1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 6 - 2
[1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 6 - 2
[1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 2
[1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 2
[1,1,0,1,1,0,0,0]
 => [[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 2
[1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 6 - 2
[1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 8 - 2
[1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 2
[1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 6 - 2
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,.],[.,.]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 10 - 2
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[.,[.,.]],[.,.]]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 2
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 10 - 2
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 2
[1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 10 - 2
[1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 2
[1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,1,0,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 10 - 2
[1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 10 - 2
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 10 - 2
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 10 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 12 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 10 - 2
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 10 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [[.,[[.,[.,.]],.]],.]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 12 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],.]],.]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 10 - 2
[1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,1,1,0,0,1,0,0,1,0]
 => [[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 2
[1,1,1,0,0,1,0,1,0,0]
 => [[[.,.],[.,[.,.]]],.]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 10 - 2
[1,1,1,0,0,1,1,0,0,0]
 => [[[.,.],[[.,.],.]],.]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 2
[1,1,1,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,.]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 12 - 2
[1,1,1,0,1,0,0,1,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 12 - 2
[1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 12 - 2
[1,1,1,0,1,1,0,0,0,0]
 => [[[.,[[.,.],.]],.],.]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 12 - 2
[1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,1,1,1,0,0,0,1,0,0]
 => [[[[.,.],.],[.,.]],.]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 10 - 2
[1,1,1,1,0,0,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 2
[1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 10 - 2
[1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 7 - 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 12 - 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,.],[.,[.,.]]]]]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,.],[[.,.],.]]]]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 18 - 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => [.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 12 - 2
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => [.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 18 - 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [.,[.,[[[.,.],[.,.]],.]]]
 => [.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 12 - 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 18 - 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,.],[.,[.,[.,.]]]]]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [.,[[.,.],[.,[[.,.],.]]]]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => [.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [.,[[.,.],[[.,[.,.]],.]]]
 => [.,[[.,[[.,.],[.,.]]],.]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 12 - 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [.,[[.,.],[[[.,.],.],.]]]
 => [.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [.,[[.,[.,.]],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 18 - 2
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [.,[[.,[.,.]],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 18 - 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => [.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? = 12 - 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => [.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 18 - 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => [.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 12 - 2
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => [.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? = 18 - 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [.,[[.,[[.,.],[.,.]]],.]]
 => [.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 0 - 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [.,[[[.,.],.],[.,[.,.]]]]
 => [.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [.,[[[.,.],.],[[.,.],.]]]
 => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => [.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [[.,.],[.,[[[.,.],.],.]]]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],[[.,.],.]]]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 8 - 2
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 8 compositions to match this statistic)
Mapping
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 40%
Values
[1,0]
 => [.,.]
 => [.,.]
 => ([],1)
 => ? = 3 - 3
[1,0,1,0]
 => [.,[.,.]]
 => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 4 - 3
[1,1,0,0]
 => [[.,.],.]
 => [[.,.],.]
 => ([(0,1)],2)
 => ? = 4 - 3
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => [.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 2 = 5 - 3
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => [.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 2 = 5 - 3
[1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => [[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 2 = 5 - 3
[1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 6 - 3
[1,1,1,0,0,0]
 => [[[.,.],.],.]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 2 = 5 - 3
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 6 - 3
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 6 - 3
[1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 6 - 3
[1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 8 - 3
[1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 6 - 3
[1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 6 - 3
[1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 6 - 3
[1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 3
[1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 3
[1,1,0,1,1,0,0,0]
 => [[.,[[.,.],.]],.]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 3
[1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 6 - 3
[1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 8 - 3
[1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 8 - 3
[1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 6 - 3
[1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,.]]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[[.,.],.]]]]
 => [.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[[.,.],[.,.]]]]
 => [.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[[.,[.,.]],.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 10 - 3
[1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,0,1,1,0,0,1,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,0,1,1,0,0,1,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,0,1,1,0,1,0,0,1,0]
 => [.,[[.,[.,.]],[.,.]]]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 3
[1,0,1,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 10 - 3
[1,0,1,1,0,1,1,0,0,0]
 => [.,[[.,[[.,.],.]],.]]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 3
[1,0,1,1,1,0,0,0,1,0]
 => [.,[[[.,.],.],[.,.]]]
 => [.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,0,1,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 10 - 3
[1,0,1,1,1,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],.]]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 3
[1,0,1,1,1,1,0,0,0,0]
 => [.,[[[[.,.],.],.],.]]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,1,0,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,1,0,0,1,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => [[.,[.,[[.,.],.]]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,1,0,0,1,1,0,0,1,0]
 => [[.,.],[[.,.],[.,.]]]
 => [[.,[[.,[.,.]],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,1,0,0,1,1,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 10 - 3
[1,1,0,0,1,1,1,0,0,0]
 => [[.,.],[[[.,.],.],.]]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,1,0,1,0,0,1,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 10 - 3
[1,1,0,1,0,0,1,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 10 - 3
[1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 10 - 3
[1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 12 - 3
[1,1,0,1,0,1,1,0,0,0]
 => [[.,[.,[[.,.],.]]],.]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 10 - 3
[1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,.]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 10 - 3
[1,1,0,1,1,0,0,1,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 - 3
[1,1,0,1,1,0,1,0,0,0]
 => [[.,[[.,[.,.]],.]],.]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 12 - 3
[1,1,0,1,1,1,0,0,0,0]
 => [[.,[[[.,.],.],.]],.]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 10 - 3
[1,1,1,0,0,0,1,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,1,1,0,0,0,1,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,1,1,0,0,1,0,0,1,0]
 => [[[.,.],[.,.]],[.,.]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 3
[1,1,1,0,0,1,0,1,0,0]
 => [[[.,.],[.,[.,.]]],.]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 10 - 3
[1,1,1,0,0,1,1,0,0,0]
 => [[[.,.],[[.,.],.]],.]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 3
[1,1,1,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,.]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 12 - 3
[1,1,1,0,1,0,0,1,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 12 - 3
[1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 12 - 3
[1,1,1,0,1,1,0,0,0,0]
 => [[[.,[[.,.],.]],.],.]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 12 - 3
[1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,1,1,1,0,0,0,1,0,0]
 => [[[[.,.],.],[.,.]],.]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 10 - 3
[1,1,1,1,0,0,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 12 - 3
[1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 10 - 3
[1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 7 - 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 12 - 3
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [.,[.,[[.,.],[.,[.,.]]]]]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [.,[.,[[.,.],[[.,.],.]]]]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 18 - 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => [.,[.,[[.,.],[.,[.,.]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 12 - 3
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => [.,[.,[[.,.],[[.,.],.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 18 - 3
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [.,[.,[[[.,.],[.,.]],.]]]
 => [.,[.,[[[.,.],[.,.]],.]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 12 - 3
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 18 - 3
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [.,[.,[[[[.,.],.],.],.]]]
 => [.,[.,[[[[.,.],.],.],.]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [.,[[.,.],[.,[.,[.,.]]]]]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [.,[[.,.],[.,[[.,.],.]]]]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => [.,[[.,[[.,[.,.]],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [.,[[.,.],[[.,[.,.]],.]]]
 => [.,[[.,[[.,.],[.,.]]],.]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 12 - 3
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [.,[[.,.],[[[.,.],.],.]]]
 => [.,[[.,[[[.,.],.],.]],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [.,[[.,[.,.]],[.,[.,.]]]]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 18 - 3
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [.,[[.,[.,.]],[[.,.],.]]]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 18 - 3
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => [.,[[.,[.,.]],[.,[.,.]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? = 12 - 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => [.,[[.,.],[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 18 - 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [.,[[.,[.,[[.,.],.]]],.]]
 => [.,[[.,.],[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 12 - 3
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => [.,[[.,[.,.]],[[.,.],.]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ? = 18 - 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [.,[[.,[[.,.],[.,.]]],.]]
 => [.,[[.,.],[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ? = 0 - 3
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [.,[[[.,.],.],[.,[.,.]]]]
 => [.,[[[.,[.,[.,.]]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [.,[[[.,.],.],[[.,.],.]]]
 => [.,[[[.,[[.,.],.]],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => [.,[[[[.,[.,.]],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[[[.,.],.],.],.],.]]
 => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,1,0,0,1,0,1,1,0,0,1,0]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [[.,.],[.,[[[.,.],.],.]]]
 => [[.,[.,[[[.,.],.],.]]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,1,0,0,1,1,0,0,1,0,1,0]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],[[.,.],.]]]
 => [[.,[[.,[[.,.],.]],.]],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 8 - 3
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001875
Mapping
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1,0]
 => [1,0]
 => ([],1)
 => ([(0,1)],2)
 => ? = 3 - 1
[1,0,1,0]
 => [1,1,0,0]
 => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 4 = 5 - 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4 = 5 - 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4 = 5 - 1
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 6 - 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 6 - 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 5 = 6 - 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 5 = 6 - 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 8 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5 = 6 - 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 5 = 6 - 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 5 = 6 - 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 8 - 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 8 - 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 8 - 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 5 = 6 - 1
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 8 - 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 8 - 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 7 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 7 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 7 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
 => ? = 10 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 7 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 7 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 7 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
 => ? = 12 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
 => ? = 10 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 12 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 7 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
 => ? = 10 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
 => ? = 12 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 6 = 7 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 7 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 7 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 7 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 10 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 6 = 7 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
 => ? = 10 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 10 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
 => ? = 10 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => ?
 => ? = 12 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 10 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 10 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
 => ? = 0 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
 => ? = 12 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 10 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 7 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 6 = 7 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
 => ? = 12 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 10 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
 => ? = 12 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
 => ? = 12 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
 => ? = 12 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([],5)
 => ?
 => ? = 12 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 12 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 6 = 7 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? = 10 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? = 12 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
 => ? = 10 - 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)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 7 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [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)
 => ([(0,5),(0,6),(1,11),(2,8),(3,7),(4,10),(5,9),(6,1),(6,9),(8,7),(9,4),(9,11),(10,3),(10,8),(11,2),(11,10)],12)
 => ? = 8 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [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)
 => ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
 => ? = 8 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,4),(0,5),(1,8),(2,10),(3,7),(4,9),(5,9),(6,3),(6,10),(7,8),(9,2),(9,6),(10,1),(10,7)],11)
 => ? = 8 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(0,6),(1,10),(2,12),(3,7),(4,9),(4,11),(5,8),(5,11),(6,8),(6,9),(7,10),(8,13),(9,13),(11,2),(11,13),(12,1),(12,7),(13,3),(13,12)],14)
 => ? = 12 - 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)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
Description
The number of simple modules with projective dimension at most 1.