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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000226
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
St000226: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000226: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => 0
[1,0,1,0]
=> [3,1,2] => 0
[1,1,0,0]
=> [2,3,1] => 3
[1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,0,1,1,0,0]
=> [3,1,4,2] => 5
[1,1,0,0,1,0]
=> [2,4,1,3] => 5
[1,1,0,1,0,0]
=> [4,3,1,2] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => 4
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 5
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 7
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 5
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 4
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 7
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 6
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 7
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 6
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 4
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 4
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 5
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 5
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 7
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 5
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 4
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 9
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 7
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 7
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 7
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 6
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 6
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 5
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 9
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 7
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 8
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 9
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 5
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 6
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 7
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 9
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 7
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 7
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 5
Description
The convexity of a permutation.
It is given by the maximal value of $2x_i-x_{i-1}-x_{i+1}$ over all $i \in \{2,\ldots,n-1\}$.
Matching statistic: St001232
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 50%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,5,5} + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {1,5,5} + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,5,5} + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,2,4,4,5,5,5,7,7,7} + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {0,2,4,4,5,5,5,7,7,7} + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,2,4,4,5,5,5,7,7,7} + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {0,2,4,4,5,5,5,7,7,7} + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {0,2,4,4,5,5,5,7,7,7} + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {0,2,4,4,5,5,5,7,7,7} + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {0,2,4,4,5,5,5,7,7,7} + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {0,2,4,4,5,5,5,7,7,7} + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {0,2,4,4,5,5,5,7,7,7} + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {0,2,4,4,5,5,5,7,7,7} + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9 = 8 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [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]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,0,1,0,1,1,0,0]
=> [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]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,0,1,1,0,0,1,0]
=> [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]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,0,1,1,0,1,0,0]
=> [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]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,0,1,1,1,0,0,0]
=> [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]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> 10 = 9 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [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]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,1,0,0,0,1,1,0,0]
=> [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]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [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]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,1,0,0,1,0,1,0,0]
=> [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]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,1,0,0,1,1,0,0,0]
=> [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]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 9 = 8 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [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]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,1,1,0,0,0,1,0,0]
=> [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]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,1,1,0,0,1,0,0,0]
=> [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]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,9,9,9,9} + 1
[1,1,1,1,1,0,0,0,0,0]
=> [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]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1 = 0 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000422
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 40%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 40%
Values
[1,0]
=> [2,1] => [1,2] => ([],2)
=> 0
[1,0,1,0]
=> [3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 3
[1,1,0,0]
=> [2,3,1] => [1,2,3] => ([],3)
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,5,5}
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,5,5}
[1,1,0,1,0,0]
=> [4,3,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,5,5}
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [3,4,6,2,1,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,5,2,6,1,4] => ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,5,2,1,4,6] => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [3,5,6,4,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [3,5,6,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [3,5,1,4,2,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,6,2,4,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,6,2,1,5,4] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [4,2,5,6,1,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [4,2,6,3,1,5] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [4,2,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [4,5,3,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [4,5,3,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,5,1,3,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,6,3,2,1,5] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [4,6,1,3,5,2] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [4,1,3,2,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [5,2,3,6,1,4] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [5,2,3,1,4,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [5,2,6,4,1,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [5,2,6,1,4,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [5,2,1,4,3,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 8
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [5,6,3,1,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [5,6,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [5,1,3,4,2,6] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6)
=> 0
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000510
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 40%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 40%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,3}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,3}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 8
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 8
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 8
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 0
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000937
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 20%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000937: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 20%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,3}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,3}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 5
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 5
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 5
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 5
Description
The number of positive values of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St001279
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 30%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 30%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,3}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,3}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,9}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 6
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 6
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 6
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 4
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 8
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St001780
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001780: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 30%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001780: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 30%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,3}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,3}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,4,4,4,5,5,5,6,6,7,7,7}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,4,4,4,4,5,5,5,5,5,5,5,5,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 6
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 6
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 6
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 6
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 9
Description
The order of promotion on the set of standard tableaux of given shape.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ? = 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> ? = 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,5,5}
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,5,5}
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,5,5}
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {0,1,5,5}
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,1,1,0,1,0,0,0]
=> [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)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,0,2,4,4,5,5,6,6,7,7,7}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[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]
=> ([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [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)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,5),(2,3),(2,4),(4,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [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)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [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)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [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)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [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)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ([(0,5),(4,3),(5,1),(5,2),(5,4)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,5),(1,2),(1,3),(1,4),(4,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,5),(2,3),(2,4),(2,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [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)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [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)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [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)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(1,5),(2,5),(3,4),(3,5)],6)
=> ? ∊ {0,1,2,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,9}
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001982
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001982: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 40%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001982: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 40%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,3}
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {0,3}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,4,5,5}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,2,4,4,5,5,5,6,6,7,7,7}
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 5
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9}
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 5
[1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 9
[1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 9
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 9
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 4
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 8
Description
The number of orbits of the action of a permutation of given cycle type on the set of edges of the complete graph.
For example, the two orbits of the action of $(1,2)(3)$ on the set $\{\{1,2\},\{1,3\},\{2,3\}\}$ are $\{\{1,2\}\}$ and $\{\{1,3\},\{2,3\}\}$.
The number of orbits of a permutation with cycle type $\mu$ is
$$
\sum_i \lfloor\frac{\mu_i}{2}\rfloor + \sum_{i < j} \gcd(\mu_i, \mu_j).
$$
The first sum counts orbits of edges that have both vertices in a single cycle of the permutation, the second counts orbits of edges with vertices in two different cycles.
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