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Your data matches 315 different statistics following compositions of up to 3 maps.
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Matching statistic: St001880
(load all 126 compositions to match this statistic)
(load all 126 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> 5
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 6
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> 6
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 6
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 7
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 7
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 5
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 7
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,4,2,5,6,3,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 6
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,2,5,3,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 7
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,4,5,2,6,3,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> 6
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,4,5,6,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 4
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> 5
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> 6
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,2,3,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> 6
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,2,5,3,6,4,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 7
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> 5
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,5,2,3,6,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 6
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,5,2,6,3,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 6
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 4
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000019
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 80%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 80%
Values
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 3 - 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 4 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 3 = 5 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 3 = 5 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? = 5 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2 = 4 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => 3 = 5 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => 4 = 6 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => 3 = 5 - 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 4 = 6 - 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5] => 4 = 6 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 4 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 3 = 5 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 4 = 6 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 3 = 5 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 6 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 4 - 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,5,6] => 3 = 5 - 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,4,6] => ? = 6 - 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6] => 4 = 6 - 2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,3,6] => ? = 6 - 2
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6] => ? = 7 - 2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 7 - 2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 6 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => 5 = 7 - 2
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,2,4,5,7,6] => 3 = 5 - 2
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 6 - 2
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,1,2,5,7,4,6] => ? = 7 - 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 7 - 2
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,2,5,7,6] => ? = 6 - 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,2,7,5,6] => ? = 7 - 2
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,1,4,5,2,7,6] => ? = 6 - 2
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2 = 4 - 2
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => 3 = 5 - 2
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => 4 = 6 - 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 4 = 6 - 2
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => 4 = 6 - 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 7 - 2
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [4,1,2,5,3,7,6] => ? = 7 - 2
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => 3 = 5 - 2
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 4 = 6 - 2
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [4,1,5,2,3,7,6] => ? = 6 - 2
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5] => 4 = 6 - 2
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 4 - 2
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 3 = 5 - 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => 5 = 7 - 2
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 4 = 6 - 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 3 = 5 - 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 4 - 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 7 - 2
Description
The cardinality of the support of a permutation.
A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$.
The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product.
See [2], Definition 1 and Proposition 10.
The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$.
Thus, the connectivity set is the complement of the support.
Matching statistic: St000067
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000067: Alternating sign matrices ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 80%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000067: Alternating sign matrices ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 80%
Values
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 3 - 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [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 = 4 - 2
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 4 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 3 = 5 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3 = 5 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [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 = 4 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 5 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [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 = 4 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 4 = 6 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 3 = 5 - 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 4 = 6 - 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4 = 6 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2 = 4 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4 = 6 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3 = 5 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [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 = 4 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 6 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 4 - 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 5 - 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 6 - 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 6 - 2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 6 - 2
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,-1,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7 - 2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 7 - 2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,-1,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 6 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> 5 = 7 - 2
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 5 - 2
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 6 - 2
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,-1,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7 - 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7 - 2
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 6 - 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7 - 2
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 6 - 2
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> [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 = 4 - 2
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 3 = 5 - 2
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 4 = 6 - 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 4 = 6 - 2
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 6 - 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 7 - 2
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 7 - 2
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 3 = 5 - 2
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> 4 = 6 - 2
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,-1,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 6 - 2
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4 = 6 - 2
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2 = 4 - 2
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 3 = 5 - 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> 5 = 7 - 2
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4 = 6 - 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3 = 5 - 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [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 = 4 - 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 7 - 2
Description
The inversion number of the alternating sign matrix.
If we denote the entries of the alternating sign matrix as $a_{i,j}$, the inversion number is defined as
$$\sum_{i > k}\sum_{j < \ell} a_{i,j}a_{k,\ell}.$$
When restricted to permutation matrices, this gives the usual inversion number of the permutation.
Matching statistic: St001480
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
[1,1,1,0,0,0]
=> []
=> []
=> ? = 3 - 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 4 - 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 5 - 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 4 = 6 - 2
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 4 = 6 - 2
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 6 - 2
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 6 - 2
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4 - 2
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5 - 2
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 6 - 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 6 - 2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 6 - 2
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 7 - 2
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7 - 2
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 6 - 2
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7 - 2
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 5 - 2
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 6 - 2
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 7 - 2
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 7 - 2
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> ? = 6 - 2
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 7 - 2
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 6 - 2
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 4 = 6 - 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 6 - 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 7 - 2
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> ? = 7 - 2
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 4 = 6 - 2
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 6 - 2
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 4 = 6 - 2
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 5 - 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 7 - 2
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 6 - 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 5 - 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 7 - 2
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
Matching statistic: St000673
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000673: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 3
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 5
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 5
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? = 5
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 4
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => 5
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => 6
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 6
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => 5
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 6
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5] => 6
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 5
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 6
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 5
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 6
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 4
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,5,6] => ? = 5
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,4,6] => ? = 6
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6] => ? = 6
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,3,6] => ? = 6
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6] => ? = 7
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 7
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 7
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,2,4,5,7,6] => ? = 5
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 6
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,1,2,5,7,4,6] => ? = 7
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 7
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,2,5,7,6] => ? = 6
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,2,7,5,6] => ? = 7
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,1,4,5,2,7,6] => ? = 6
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 4
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => 5
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => 6
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 6
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 6
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 7
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [4,1,2,5,3,7,6] => ? = 7
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => 5
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 6
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [4,1,5,2,3,7,6] => ? = 6
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5] => 6
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 5
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 7
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 6
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 5
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 7
Description
The number of non-fixed points of a permutation.
In other words, this statistic is $n$ minus the number of fixed points ([[St000022]]) of $\pi$.
Matching statistic: St001005
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001005: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 3
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 5
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 5
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? = 5
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 4
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => 5
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => 6
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 6
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => 5
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 6
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5] => 6
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 5
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 6
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 5
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 6
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 4
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,5,6] => ? = 5
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,4,6] => ? = 6
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6] => ? = 6
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,3,6] => ? = 6
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6] => ? = 7
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 7
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 7
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,2,4,5,7,6] => ? = 5
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 6
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,1,2,5,7,4,6] => ? = 7
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 7
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,2,5,7,6] => ? = 6
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,2,7,5,6] => ? = 7
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,1,4,5,2,7,6] => ? = 6
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 4
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => 5
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => 6
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 6
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 6
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 7
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [4,1,2,5,3,7,6] => ? = 7
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => 5
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 6
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [4,1,5,2,3,7,6] => ? = 6
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5] => 6
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 5
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 7
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 6
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 5
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 7
Description
The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.
Matching statistic: St001077
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001077: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001077: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 3
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 4
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 5
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 5
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? = 5
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 4
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => 5
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => 6
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 6
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => 5
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 6
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5] => 6
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 5
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 6
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 5
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 6
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 4
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,5,6] => ? = 5
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,4,6] => ? = 6
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6] => ? = 6
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,3,6] => ? = 6
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,3,5,6] => ? = 7
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,3,4,7,6] => ? = 7
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,3,4,6] => ? = 6
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 7
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,2,4,5,7,6] => ? = 5
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 6
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,1,2,5,7,4,6] => ? = 7
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 7
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [3,1,4,2,5,7,6] => ? = 6
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,1,4,2,7,5,6] => ? = 7
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,1,4,5,2,7,6] => ? = 6
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 4
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,4,5] => 5
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5] => 6
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,3,4,5] => 6
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 6
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 7
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [4,1,2,5,3,7,6] => ? = 7
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2,4,6,5] => 5
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,6,4,5] => 6
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [4,1,5,2,3,7,6] => ? = 6
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,4,2,6,5] => 6
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 5
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 7
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5] => 6
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 5
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 7
Description
The prefix exchange distance of a permutation.
This is the number of star transpositions needed to write a permutation.
In symbols, for a permutation $\pi\in\mathfrak S_n$ this is
$$\min\{ k \mid \pi = \tau_{i_1} \cdots \tau_{i_k}, 2 \leq i_1,\ldots,i_k \leq n\},$$
where $\tau_a = (1,a)$ for $2 \leq a \leq n$.
[1, Lem. 2.1] shows that the this length is $n+m-a-1$, where $m$ is the number of non-trival cycles not containing the element $1$, and $a$ is the number of fixed points different from $1$.
One may find in [2] explicit formulas for its generating function and a combinatorial proof that it is asymptotically normal.
Matching statistic: St000235
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 4 = 5 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 4 = 5 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? = 5 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => 5 = 6 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => 5 = 6 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => 4 = 5 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => 5 = 6 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,2,5,3,6] => 5 = 6 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 4 = 5 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => 5 = 6 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 4 = 5 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 6 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => ? = 4 - 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,2,6,7] => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5,7] => ? = 6 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => ? = 6 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,6,2,4,7] => ? = 6 - 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,5,2,4,6,7] => ? = 7 - 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,4,6,5,7] => ? = 7 - 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,6,2,4,5,7] => ? = 6 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => ? = 7 - 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,5,6,3,7] => ? = 5 - 1
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => ? = 6 - 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5,7] => ? = 7 - 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => ? = 7 - 1
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,4,2,5,6,3,7] => ? = 6 - 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,2,5,3,6,7] => ? = 7 - 1
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,4,5,2,6,3,7] => ? = 6 - 1
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => 3 = 4 - 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 4 = 5 - 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => 5 = 6 - 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => 5 = 6 - 1
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => ? = 6 - 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,6,7] => ? = 7 - 1
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,2,5,3,6,4,7] => ? = 7 - 1
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => 4 = 5 - 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => 5 = 6 - 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,5,2,3,6,4,7] => ? = 6 - 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,2,5,3,6] => 5 = 6 - 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 4 = 5 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => ? = 7 - 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => 5 = 6 - 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 4 = 5 - 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 7 - 1
Description
The number of indices that are not cyclical small weak excedances.
A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.
Matching statistic: St000240
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000240: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000240: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
[1,1,1,0,0,0]
=> []
=> []
=> [] => ? = 3 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> [] => ? = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 4 = 5 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 4 = 5 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> [] => ? = 5 - 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => 3 = 4 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => 5 = 6 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => 5 = 6 - 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => 4 = 5 - 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => 5 = 6 - 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,2,5,3,6] => 5 = 6 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 4 = 5 - 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => 5 = 6 - 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 4 = 5 - 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 6 - 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,2,7] => ? = 4 - 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,3,4,5,2,6,7] => ? = 5 - 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5,7] => ? = 6 - 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => ? = 6 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,6,2,4,7] => ? = 6 - 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,5,2,4,6,7] => ? = 7 - 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,4,6,5,7] => ? = 7 - 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,6,2,4,5,7] => ? = 6 - 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,4,5,6,7] => ? = 7 - 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,5,6,3,7] => ? = 5 - 1
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => ? = 6 - 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5,7] => ? = 7 - 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,2,4,3,5,6,7] => ? = 7 - 1
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,4,2,5,6,3,7] => ? = 6 - 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,2,5,3,6,7] => ? = 7 - 1
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,4,5,2,6,3,7] => ? = 6 - 1
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => 3 = 4 - 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 4 = 5 - 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4,6] => 5 = 6 - 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => 5 = 6 - 1
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => ? = 6 - 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,6,7] => ? = 7 - 1
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,2,5,3,6,4,7] => ? = 7 - 1
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => 4 = 5 - 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,4,3,5,6] => 5 = 6 - 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,5,2,3,6,4,7] => ? = 6 - 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,2,5,3,6] => 5 = 6 - 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 3 = 4 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 4 = 5 - 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,2,3,4,6,5,7] => ? = 7 - 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => 5 = 6 - 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 4 = 5 - 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 3 = 4 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> [] => ? = 7 - 1
Description
The number of indices that are not small excedances.
A small excedance is an index $i$ for which $\pi_i = i+1$.
Matching statistic: St001255
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001255: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001255: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
[1,1,1,0,0,0]
=> []
=> []
=> []
=> ? = 3 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 5 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 7 = 6 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 6 + 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 4 + 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 5 + 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2]
=> [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 6 + 1
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 6 + 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 6 + 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 7 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [5,5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 7 + 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 6 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 7 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 5 + 1
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 6 + 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 7 + 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 7 + 1
[1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 7 + 1
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 7 + 1
[1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 7 + 1
[1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 6 + 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 7 = 6 + 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 7 + 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5 = 4 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> []
=> ? = 7 + 1
Description
The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.
The following 305 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000029The depth of a permutation. St000216The absolute length of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of 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. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000144The pyramid weight of the Dyck path. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000907The number of maximal antichains of minimal length in a poset. St001090The number of pop-stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000837The number of ascents of distance 2 of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000702The number of weak deficiencies of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000308The height of the tree associated to a permutation. St000746The number of pairs with odd minimum in a perfect matching. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001589The nesting number of a perfect matching. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001520The number of strict 3-descents. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000141The maximum drop size of a permutation. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St000366The number of double descents of a permutation. St001596The number of two-by-two squares inside a skew partition. St000044The number of vertices of the unicellular map given by a perfect matching. St000422The energy of a graph, if it is integral. St000643The size of the largest orbit of antichains under Panyushev complementation. St000653The last descent of a permutation. St000863The length of the first row of the shifted shape of a permutation. St001391The disjunction number of a graph. St001649The length of a longest trail in a graph. St001782The order of rowmotion on the set of order ideals of a poset. St000528The height of a poset. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000906The length of the shortest maximal chain in a poset. St000996The number of exclusive left-to-right maxima of a permutation. St001246The maximal difference between two consecutive entries 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. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000080The rank of the poset. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000454The largest eigenvalue of a graph if it is integral. St000731The number of double exceedences of a permutation. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001510The number of self-evacuating linear extensions of a finite poset. St001517The length of a longest pair of twins in a permutation. St001566The length of the longest arithmetic progression in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001902The number of potential covers of a poset. St000358The number of occurrences of the pattern 31-2. St000624The normalized sum of the minimal distances to a greater element. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St000912The number of maximal antichains in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001245The cyclic maximal difference between two consecutive entries of a permutation. St000956The maximal displacement of a permutation. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001343The dimension of the reduced incidence algebra of a poset. 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. St000028The number of stack-sorts needed to sort a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000356The number of occurrences of the pattern 13-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St000443The number of long tunnels of a Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000039The number of crossings of a permutation. St000015The number of peaks of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000060The greater neighbor of the maximum. St000064The number of one-box pattern of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000209Maximum difference of elements in cycles. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000519The largest length of a factor maximising the subword complexity. St000619The number of cyclic descents of a permutation. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000922The minimal number such that all substrings of this length are unique. St000991The number of right-to-left minima of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001345The Hamming dimension of a graph. St001375The pancake length of a permutation. St001497The position of the largest weak excedence of a permutation. St001512The minimum rank of a graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St000021The number of descents of a permutation. St000056The decomposition (or block) number of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000222The number of alignments in the permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000354The number of recoils of a permutation. St000539The number of odd inversions of a permutation. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000829The Ulam distance of a permutation to the identity permutation. St000836The number of descents of distance 2 of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001093The detour number of a graph. 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. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001285The number of primes in the column sums of the two line notation of a permutation. St001366The maximal multiplicity of a degree of a vertex of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001430The number of positive entries in a signed permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001668The number of points of the poset minus the width of the poset. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001875The number of simple modules with projective dimension at most 1. St000241The number of cyclical small excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000317The cycle descent number of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000637The length of the longest cycle in a graph. St000711The number of big exceedences of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000989The number of final rises of a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension 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 $eA$. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001557The number of inversions of the second entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000365The number of double ascents of a permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001394The genus of a permutation. St001524The degree of symmetry of a binary word. St001556The number of inversions of the third entry of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001742The difference of the maximal and the minimal degree in a graph. St000455The second largest eigenvalue of a graph if it is integral. St001488The number of corners of a skew partition. St001645The pebbling number of a connected graph. St000018The number of inversions of a permutation. St000463The number of admissible inversions of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001487The number of inner corners of a skew partition. St001812The biclique partition number of a graph. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St001323The independence gap of a graph. St001435The number of missing boxes in the first row. St001490The number of connected components of a skew partition. St001577The minimal number of edges to add or remove to make a graph a cograph. St000381The largest part of an integer composition. St001820The size of the image of the pop stack sorting operator. St000161The sum of the sizes of the right subtrees of a binary tree. St000327The number of cover relations in a poset. St001330The hat guessing number of a graph. St001637The number of (upper) dissectors of a poset. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000292The number of ascents of a binary word. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001925The minimal number of zeros in a row of an alternating sign matrix. St001964The interval resolution global dimension of a poset. St000006The dinv of a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000741The Colin de Verdière graph invariant. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St000911The number of maximal antichains of maximal size in a poset. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001052The length of the exterior of a permutation. St000390The number of runs of ones in a binary word. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000924The number of topologically connected components of a perfect matching. St001115The number of even descents of a permutation. St001834The number of non-isomorphic minors of a graph. St000291The number of descents of a binary word. St001769The reflection length of a signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001866The nesting alignments of a signed permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph.
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