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Your data matches 163 different statistics following compositions of up to 3 maps.
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Matching statistic: St001343
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001343: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001343: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 2
[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]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 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]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 7
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 7
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 7
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 7
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[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]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[]
=> [1,0]
=> [1,0]
=> ([],1)
=> 1
Description
The dimension of the reduced incidence algebra of a poset.
The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets.
Thus, this statistic returns the number of non-isomorphic intervals of the poset.
Matching statistic: St001645
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 86%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 3 - 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 3 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? = 4 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 5 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 7 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 7 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 7 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,1,3] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,1,3] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,1,3] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 7 - 1
[]
=> [] => [] => ([],0)
=> ? = 1 - 1
Description
The pebbling number of a connected graph.
Matching statistic: St001603
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 65%●distinct values known / distinct values provided: 29%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 65%●distinct values known / distinct values provided: 29%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 2 - 5
[1,0,1,0]
=> [2,1] => [2]
=> []
=> ? = 3 - 5
[1,1,0,0]
=> [1,2] => [1,1]
=> [1]
=> ? = 3 - 5
[1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? = 4 - 5
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? = 4 - 5
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? = 4 - 5
[1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? = 4 - 5
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> ? = 5 - 5
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> ? = 5 - 5
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> [1,1]
=> ? = 5 - 5
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> ? = 5 - 5
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> ? = 5 - 5
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> ? = 7 - 5
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,1]
=> ? = 7 - 5
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [2,1,1]
=> [1,1]
=> ? = 7 - 5
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> ? = 7 - 5
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [2,2,1]
=> [2,1]
=> 1 = 6 - 5
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> 1 = 6 - 5
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [2,2,2]
=> [2,2]
=> 2 = 7 - 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [2,2,2]
=> [2,2]
=> 2 = 7 - 5
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5] => [2,2,2]
=> [2,2]
=> 2 = 7 - 5
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,4,1,2,6,5] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [2,2,2]
=> [2,2]
=> 2 = 7 - 5
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [2,2,1,1]
=> [2,1,1]
=> 2 = 7 - 5
[]
=> [] => []
=> ?
=> ? = 1 - 5
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001232
(load all 170 compositions to match this statistic)
(load all 170 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 3 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 4 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 4 - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 5 - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? = 7 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? = 7 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 7 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 7 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001604
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 33%●distinct values known / distinct values provided: 14%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 33%●distinct values known / distinct values provided: 14%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 2 - 6
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? = 3 - 6
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? = 3 - 6
[1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? = 4 - 6
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? = 4 - 6
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? = 4 - 6
[1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? = 4 - 6
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? = 5 - 6
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> ? = 5 - 6
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> ? = 5 - 6
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> ? = 5 - 6
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> ? = 5 - 6
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? = 7 - 6
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? = 7 - 6
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? = 7 - 6
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? = 7 - 6
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,4,1,2,6,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[]
=> [] => []
=> ?
=> ? = 1 - 6
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001605
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 33%●distinct values known / distinct values provided: 14%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 33%●distinct values known / distinct values provided: 14%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 2 - 6
[1,0,1,0]
=> [2,1] => [1,1]
=> [1]
=> ? = 3 - 6
[1,1,0,0]
=> [1,2] => [2]
=> []
=> ? = 3 - 6
[1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? = 4 - 6
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? = 4 - 6
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? = 4 - 6
[1,1,0,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> ? = 4 - 6
[1,1,1,0,0,0]
=> [1,2,3] => [3]
=> []
=> ? = 5 - 6
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> ? = 5 - 6
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,2]
=> [2]
=> ? = 5 - 6
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> [2]
=> ? = 5 - 6
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> ? = 5 - 6
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [3,1]
=> [1]
=> ? = 7 - 6
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> ? = 7 - 6
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> ? = 7 - 6
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [3,1]
=> [1]
=> ? = 7 - 6
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [3,2]
=> [2]
=> ? = 6 - 6
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,5,1,6,3,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,4,1,2,6,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [3,4,1,6,2,5] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [3,1,5,2,6,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,5,1,2,6,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,1,5,6,2,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,5,1,6,2,4] => [3,3]
=> [3]
=> 1 = 7 - 6
[]
=> [] => []
=> ?
=> ? = 1 - 6
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001875
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
[1,0]
=> ([],1)
=> ([(0,1)],2)
=> ? = 2
[1,0,1,0]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0]
=> ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[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
[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)
=> ? = 5
[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
[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
[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
[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
[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)
=> ? = 7
[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)
=> ? = 7
[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)
=> ? = 7
[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)
=> ? = 7
[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
[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
[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)
=> ? = 6
[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)
=> ? = 6
[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
[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)
=> ? = 6
[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)
=> ? = 6
[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)
=> ? = 6
[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)
=> ? = 6
[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)
=> ? = 6
[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)
=> ? = 6
[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)
=> ? = 6
[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)
=> ? = 6
[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)
=> ? = 6
[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)
=> ? = 6
[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)
=> ? = 6
[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
[1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 7
[1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 7
[1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 7
[1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
=> ? = 7
[1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 7
[1,0,1,1,0,0,1,1,0,1,0,0]
=> ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,5),(0,6),(1,8),(2,8),(4,9),(5,7),(6,4),(6,7),(7,9),(8,3),(9,1),(9,2)],10)
=> ? = 7
[1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? = 7
[1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> ? = 7
[1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> ? = 7
[1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> ([(0,4),(0,5),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,2),(6,3),(8,9),(9,6)],10)
=> ? = 7
[1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 7
[1,1,0,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
=> ? = 7
[1,1,0,1,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 7
[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)
=> ([(0,4),(0,6),(1,10),(2,9),(3,8),(4,7),(5,2),(5,8),(6,1),(6,7),(7,10),(8,9),(10,3),(10,5)],11)
=> ? = 7
[]
=> ?
=> ?
=> ? = 1
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St000337
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 86%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 86%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [4,3,1,2] => 2 = 3 - 1
[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 = 4 - 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,6,5,3,4] => 3 = 4 - 1
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [4,3,1,2,6,5] => 3 = 4 - 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [6,3,1,5,2,4] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [6,5,4,1,2,3] => 4 = 5 - 1
[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 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,8,7,5,6] => ? = 5 - 1
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [4,3,1,2,6,5,8,7] => ? = 5 - 1
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [4,3,1,2,8,7,5,6] => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [8,3,1,5,2,7,4,6] => ? = 7 - 1
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [8,3,1,7,6,2,4,5] => ? = 7 - 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [8,5,4,1,2,7,3,6] => ? = 7 - 1
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [8,7,4,1,6,2,3,5] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => 5 = 6 - 1
[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] => [2,1,4,3,8,7,5,6,10,9] => ? = 6 - 1
[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] => [2,1,4,3,10,7,5,9,6,8] => ? = 6 - 1
[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] => [2,1,4,3,10,9,8,5,6,7] => ? = 6 - 1
[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] => [2,1,6,5,3,4,8,7,10,9] => ? = 6 - 1
[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] => [2,1,8,5,3,7,4,6,10,9] => ? = 6 - 1
[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] => [2,1,10,5,3,7,4,9,6,8] => ? = 6 - 1
[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] => [2,1,10,5,3,9,8,4,6,7] => ? = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [6,3,1,5,2,4,8,7,10,9] => ? = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [8,3,1,5,2,7,4,6,10,9] => ? = 6 - 1
[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] => [10,3,1,5,2,7,4,9,6,8] => ? = 6 - 1
[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] => [10,3,1,5,2,9,8,4,6,7] => ? = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [6,5,4,1,2,3,8,7,10,9] => ? = 6 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [8,5,4,1,2,7,3,6,10,9] => ? = 6 - 1
[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] => [10,5,4,1,2,7,3,9,6,8] => ? = 6 - 1
[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] => [10,5,4,1,2,9,8,3,6,7] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => 6 = 7 - 1
[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] => [2,1,4,3,6,5,8,7,12,11,9,10] => ? = 7 - 1
[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] => [2,1,4,3,6,5,10,9,7,8,12,11] => ? = 7 - 1
[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] => [2,1,4,3,6,5,12,9,7,11,8,10] => ? = 7 - 1
[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] => [2,1,6,5,3,4,8,7,10,9,12,11] => ? = 7 - 1
[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] => [2,1,6,5,3,4,8,7,12,11,9,10] => ? = 7 - 1
[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] => [2,1,6,5,3,4,10,9,7,8,12,11] => ? = 7 - 1
[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] => [2,1,6,5,3,4,12,9,7,11,8,10] => ? = 7 - 1
[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] => [4,3,1,2,6,5,8,7,10,9,12,11] => ? = 7 - 1
[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] => [4,3,1,2,6,5,8,7,12,11,9,10] => ? = 7 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> [3,4,2,1,6,5,9,10,8,7,12,11] => [4,3,1,2,6,5,10,9,7,8,12,11] => ? = 7 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
=> [3,4,2,1,6,5,9,11,8,12,10,7] => [4,3,1,2,6,5,12,9,7,11,8,10] => ? = 7 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> [3,5,2,6,4,1,8,7,10,9,12,11] => [6,3,1,5,2,4,8,7,10,9,12,11] => ? = 7 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)]
=> [3,5,2,6,4,1,8,7,11,12,10,9] => [6,3,1,5,2,4,8,7,12,11,9,10] => ? = 7 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)]
=> [3,5,2,6,4,1,9,10,8,7,12,11] => [6,3,1,5,2,4,10,9,7,8,12,11] => ? = 7 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]
=> [3,5,2,6,4,1,9,11,8,12,10,7] => [6,3,1,5,2,4,12,9,7,11,8,10] => ? = 7 - 1
[]
=> []
=> ? => ? => ? = 1 - 1
Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation.
For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
Matching statistic: St000670
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 57%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000670: Permutations ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [2,4,3,1] => 2 = 3 - 1
[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 = 4 - 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,4,6,5,3] => 3 = 4 - 1
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [2,4,3,1,6,5] => 3 = 4 - 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [4,6,3,5,1,2] => 3 = 4 - 1
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [2,3,6,4,5,1] => 4 = 5 - 1
[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 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,6,8,7,5] => 4 = 5 - 1
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [2,4,3,1,6,5,8,7] => 4 = 5 - 1
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [2,4,3,1,6,8,7,5] => 4 = 5 - 1
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,5,3,8,2,7,1,4] => ? = 7 - 1
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [4,5,3,8,6,2,7,1] => ? = 7 - 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [6,3,8,4,5,7,1,2] => ? = 7 - 1
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [2,5,8,4,7,6,1,3] => ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [2,1,4,3,6,5,8,7,10,9] => ? = 6 - 1
[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] => [2,1,4,3,6,8,7,5,10,9] => ? = 6 - 1
[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] => [2,1,4,3,8,10,7,9,5,6] => ? = 6 - 1
[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] => [2,1,4,3,6,7,10,8,9,5] => ? = 6 - 1
[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] => [2,1,4,6,5,3,8,7,10,9] => ? = 6 - 1
[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] => [2,1,6,8,5,7,3,4,10,9] => ? = 6 - 1
[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] => [2,1,8,7,5,10,4,9,3,6] => ? = 6 - 1
[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] => [2,1,6,7,5,10,8,4,9,3] => ? = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [3,5,2,6,4,1,8,7,10,9] => [4,6,3,5,1,2,8,7,10,9] => ? = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => [6,5,3,8,2,7,1,4,10,9] => ? = 6 - 1
[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] => [8,5,3,7,2,10,4,9,1,6] => ? = 6 - 1
[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] => [6,5,3,7,2,10,8,4,9,1] => ? = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => [2,3,6,4,5,1,8,7,10,9] => ? = 6 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => [6,3,8,4,5,7,1,2,10,9] => ? = 6 - 1
[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] => [8,3,7,4,5,10,2,9,1,6] => ? = 6 - 1
[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] => [6,3,7,4,5,10,8,2,9,1] => ? = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 7 - 1
[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] => [2,1,4,3,6,5,8,7,10,12,11,9] => ? = 7 - 1
[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] => [2,1,4,3,6,5,8,10,9,7,12,11] => ? = 7 - 1
[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] => [2,1,4,3,6,5,10,12,9,11,7,8] => ? = 7 - 1
[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] => [2,1,4,6,5,3,8,7,10,9,12,11] => ? = 7 - 1
[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] => [2,1,4,6,5,3,8,7,10,12,11,9] => ? = 7 - 1
[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] => [2,1,4,6,5,3,8,10,9,7,12,11] => ? = 7 - 1
[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] => [2,1,4,6,5,3,10,12,9,11,7,8] => ? = 7 - 1
[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] => [2,4,3,1,6,5,8,7,10,9,12,11] => ? = 7 - 1
[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] => [2,4,3,1,6,5,8,7,10,12,11,9] => ? = 7 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> [3,4,2,1,6,5,9,10,8,7,12,11] => [2,4,3,1,6,5,8,10,9,7,12,11] => ? = 7 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,9),(10,11)]
=> [3,4,2,1,6,5,9,11,8,12,10,7] => [2,4,3,1,6,5,10,12,9,11,7,8] => ? = 7 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10),(11,12)]
=> [3,5,2,6,4,1,8,7,10,9,12,11] => [4,6,3,5,1,2,8,7,10,9,12,11] => ? = 7 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11)]
=> [3,5,2,6,4,1,8,7,11,12,10,9] => [4,6,3,5,1,2,8,7,10,12,11,9] => ? = 7 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12)]
=> [3,5,2,6,4,1,9,10,8,7,12,11] => [4,6,3,5,1,2,8,10,9,7,12,11] => ? = 7 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11)]
=> [3,5,2,6,4,1,9,11,8,12,10,7] => [4,6,3,5,1,2,10,12,9,11,7,8] => ? = 7 - 1
[]
=> []
=> ? => ? => ? = 1 - 1
Description
The reversal length of a permutation.
A reversal in a permutation $\pi = [\pi_1,\ldots,\pi_n]$ is a reversal of a subsequence of the form $\operatorname{reversal}_{i,j}(\pi) = [\pi_1,\ldots,\pi_{i-1},\pi_j,\pi_{j-1},\ldots,\pi_{i+1},\pi_i,\pi_{j+1},\ldots,\pi_n]$ for $1 \leq i < j \leq n$.
This statistic is then given by the minimal number of reversals needed to sort a permutation.
The reversal distance between two permutations plays an important role in studying DNA structures.
Matching statistic: St001880
(load all 61 compositions to match this statistic)
(load all 61 compositions to match this statistic)
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 57%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 57%
Values
[1,0]
=> [[1]]
=> [1] => ([],1)
=> ? = 2 - 1
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => ([(0,1)],2)
=> ? = 3 - 1
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => ([],2)
=> ? = 3 - 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ? = 4 - 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 4 - 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ? = 4 - 1
[1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 5 - 1
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? = 5 - 1
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? = 5 - 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? = 5 - 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? = 7 - 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? = 7 - 1
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? = 7 - 1
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? = 7 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 6 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? = 6 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5 = 6 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 6 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? = 6 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 6 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ? = 6 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ? = 6 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 6 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ? = 6 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? = 6 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? = 6 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 7 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6 = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> ? = 7 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6 = 7 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ? = 7 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 7 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ? = 7 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 7 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ? = 7 - 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ? = 7 - 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ? = 7 - 1
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6 = 7 - 1
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ? = 7 - 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 7 - 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ? = 7 - 1
[]
=> []
=> ? => ?
=> ? = 1 - 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
The following 153 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000703The number of deficiencies of a permutation. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001554The number of distinct nonempty subtrees of a binary tree. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St001821The sorting index of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St000863The length of the first row of the shifted shape of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000155The number of exceedances (also excedences) of a permutation. St000238The number of indices that are not small weak excedances. St000308The height of the tree associated to a permutation. St000331The number of upper interactions of a Dyck path. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001405The number of bonds in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St000133The "bounce" of a permutation. St000624The normalized sum of the minimal distances to a greater element. St001565The number of arithmetic progressions of length 2 in a permutation. St001621The number of atoms of a lattice. St000527The width of the poset. St000845The maximal number of elements covered by an element in a poset. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset. St000454The largest eigenvalue of a graph if it is integral. St000632The jump number of the poset. St000702The number of weak deficiencies of a permutation. St000462The major index minus the number of excedences of a permutation. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001812The biclique partition number of a graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000306The bounce count of a Dyck path. St001879The 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. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000004The major index of a permutation. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000075The orbit size of a standard tableau under promotion. St000105The number of blocks in the set partition. St000141The maximum drop size of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000210Minimum over maximum difference of elements in cycles. St000211The rank of the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000251The number of nonsingleton blocks of a set partition. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000443The number of long tunnels of a Dyck path. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000553The number of blocks of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000619The number of cyclic descents of a permutation. St000653The last descent of a permutation. St000654The first descent of a permutation. St000662The staircase size of the code of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000794The mak of a permutation. St000798The makl of a permutation. St000809The reduced reflection length of the permutation. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000833The comajor index of a permutation. St000925The number of topologically connected components of a set partition. St000956The maximal displacement of a permutation. St000991The number of right-to-left minima of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001220The width of a permutation. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001517The length of a longest pair of twins in a permutation. St001642The Prague dimension of a graph. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001769The reflection length of a signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001874Lusztig's a-function for the symmetric group. St000094The depth of an ordered tree. St000135The number of lucky cars of the parking function. St000168The number of internal nodes of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000325The width of the tree associated to a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000470The number of runs in a permutation. St000521The number of distinct subtrees of an ordered tree. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000744The length of the path to the largest entry in a standard Young tableau. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000989The number of final rises of a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St001417The length of a longest palindromic subword of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001596The number of two-by-two squares inside a skew partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000264The girth of a graph, which is not a tree. St000735The last entry on the main diagonal of a standard tableau.
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