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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
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(load all 25 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [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
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [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]
=> 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 5
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 6
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 6
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 6
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 6
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> 7
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 8
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 7
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 7
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> 10
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> 8
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> 8
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 6
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 9
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 8
[2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9
[6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
[4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 7
[6,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> 8
[6,4,3]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> 9
[6,4,2,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> 7
[6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 9
[6,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> 7
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000912
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000912: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 3 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 4 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 3 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [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 = 6 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 4 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [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 = 6 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5 = 4 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [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 = 6 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 6 = 5 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 7 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 8 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 5 + 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 5 + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 6 + 1
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 7 + 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 7 + 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7 + 1
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 10 + 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 8 + 1
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 8 + 1
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 6 + 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 9 + 1
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 8 + 1
[2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? = 6 + 1
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 9 + 1
[6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 6 + 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 8 + 1
[4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 7 + 1
[6,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 8 + 1
[6,4,3]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 9 + 1
[6,4,2,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ([(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
[6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 9 + 1
[6,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 7 + 1
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ([(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)
=> ? = 10 + 1
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,10),(3,7),(4,8),(5,9),(6,1),(6,9),(7,8),(8,2),(9,3),(9,10),(10,4),(10,7)],11)
=> ? = 8 + 1
[6,3,3,2]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 10 + 1
[5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 12 + 1
[3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 10 + 1
[6,5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,10),(3,7),(4,9),(5,9),(6,3),(6,10),(7,8),(9,2),(9,6),(10,1),(10,7)],11)
=> ? = 7 + 1
[5,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ([(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)
=> ? = 10 + 1
[4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 12 + 1
[6,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,6),(1,12),(2,8),(3,10),(4,11),(5,1),(5,7),(6,5),(6,11),(7,10),(7,12),(9,8),(10,9),(11,3),(11,7),(12,2),(12,9)],13)
=> ? = 10 + 1
[6,5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? = 11 + 1
[5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,6),(1,9),(2,10),(3,8),(4,7),(5,2),(5,11),(6,5),(6,7),(7,11),(8,9),(10,1),(10,8),(11,3),(11,10)],12)
=> ? = 8 + 1
[4,4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
=> ? = 9 + 1
[5,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,6),(1,12),(2,8),(3,10),(4,11),(5,3),(5,7),(6,5),(6,11),(7,10),(7,12),(9,8),(10,9),(11,1),(11,7),(12,2),(12,9)],13)
=> ? = 11 + 1
Description
The number of maximal antichains in a poset.
Matching statistic: St001000
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 50%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [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
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [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]
=> ? = 6
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 5
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 6
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 6
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 6
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 7
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 8
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 5
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 5
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 6
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 7
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 7
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 7
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 10
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 6
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 8
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 8
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 6
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 9
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 8
[2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 6
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 9
[6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 8
[4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 7
[6,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 8
[6,4,3]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 9
[6,4,2,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 7
[6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 9
[6,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 7
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 10
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 8
[6,3,3,2]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 10
[5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 12
[3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 10
[6,5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> ? = 7
[5,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 10
[4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 12
[6,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 10
[6,5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 11
[5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 8
[4,4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 9
[5,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 11
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000680
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000680: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 42%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000680: Posets ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 42%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 3 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 4 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 3 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [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)
=> ? = 6 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 4 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [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)
=> ? = 6 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5 = 4 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [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)
=> ? = 6 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 6 = 5 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? = 6 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 5 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 6 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 6 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 7 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 8 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 5 + 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 5 + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 6 + 1
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 7 + 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? = 7 + 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 7 + 1
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 10 + 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6 + 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 8 + 1
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 8 + 1
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? = 6 + 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 9 + 1
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? = 8 + 1
[2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? = 6 + 1
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 9 + 1
[6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 6 + 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
=> ? = 8 + 1
[4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? = 7 + 1
[6,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 8 + 1
[6,4,3]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 9 + 1
[6,4,2,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ([(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
[6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 9 + 1
[6,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,8),(2,9),(3,7),(4,3),(4,9),(5,6),(6,2),(6,4),(7,8),(9,1),(9,7)],10)
=> ? = 7 + 1
[6,5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ([(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)
=> ? = 10 + 1
[6,4,3,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ([(0,5),(0,6),(1,10),(3,7),(4,8),(5,9),(6,1),(6,9),(7,8),(8,2),(9,3),(9,10),(10,4),(10,7)],11)
=> ? = 8 + 1
[6,3,3,2]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ? = 10 + 1
[5,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 12 + 1
[3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 10 + 1
[6,5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,10),(3,7),(4,9),(5,9),(6,3),(6,10),(7,8),(9,2),(9,6),(10,1),(10,7)],11)
=> ? = 7 + 1
[5,5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ([(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)
=> ? = 10 + 1
[4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? = 12 + 1
[6,5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,6),(1,12),(2,8),(3,10),(4,11),(5,1),(5,7),(6,5),(6,11),(7,10),(7,12),(9,8),(10,9),(11,3),(11,7),(12,2),(12,9)],13)
=> ? = 10 + 1
[6,5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(0,6),(1,11),(2,8),(3,9),(4,3),(4,7),(5,1),(5,7),(6,4),(6,5),(7,9),(7,11),(9,10),(10,8),(11,2),(11,10)],12)
=> ? = 11 + 1
[5,5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,6),(1,9),(2,10),(3,8),(4,7),(5,2),(5,11),(6,5),(6,7),(7,11),(8,9),(10,1),(10,8),(11,3),(11,10)],12)
=> ? = 8 + 1
[4,4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,6),(1,8),(2,9),(3,10),(4,7),(5,3),(5,9),(6,2),(6,5),(7,8),(9,4),(9,10),(10,1),(10,7)],11)
=> ? = 9 + 1
[5,5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> ([(0,4),(0,6),(1,12),(2,8),(3,10),(4,11),(5,3),(5,7),(6,5),(6,11),(7,10),(7,12),(9,8),(10,9),(11,1),(11,7),(12,2),(12,9)],13)
=> ? = 11 + 1
Description
The Grundy value for Hackendot on posets.
Two players take turns and remove an order filter. The player who is faced with the one element poset looses. This game is a slight variation of Chomp.
This statistic is the Grundy value of the poset, that is, the smallest non-negative integer which does not occur as value of a poset obtained by a single move.
Matching statistic: St000062
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000062: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 3
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [1,3,2] => 2
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 4
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => 3
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,3,4,5,2] => 4
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => 4
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => 3
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 5
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => 4
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => ? = 6
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => ? = 5
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => ? = 6
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [1,3,6,7,2,5,4] => ? = 5
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [1,3,2,5,4,7,6] => ? = 4
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,2,5,6,7,8,3,4] => ? = 6
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,3,7,8,4,5,6] => ? = 5
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,3,6,7,8,2,5,4] => ? = 6
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,3,4,8,2,6,7,5] => ? = 5
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,2,5,3,4,8,6,7] => ? = 6
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,3,7,8,9,4,5,6] => ? = 6
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [1,3,6,7,8,9,2,5,4] => ? = 7
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,4,9,5,6,7,8] => ? = 8
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [1,3,4,8,9,2,6,7,5] => ? = 5
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [1,3,2,5,4,7,6,9,8] => ? = 5
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [1,2,3,4,9,10,5,6,7,8] => ? = 6
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [1,3,4,8,9,10,2,6,7,5] => ? = 6
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5,10] => [1,3,4,5,10,2,7,8,9,6] => ? = 7
[5,3,2]
=> [[1,2,5,9,10],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9,10] => [1,2,5,9,10,3,4,8,6,7] => ? = 7
[5,2,2,1]
=> [[1,3,8,9,10],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9,10] => [1,3,8,9,10,2,5,4,7,6] => ? = 7
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? => ? = 10
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? => ? = 6
[6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? => ? = 8
[6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? => ? => ? = 8
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => ? => ? = 6
[4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? => ? = 9
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? => ? = 8
[2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? => ? = 6
[6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? => ? => ? = 9
[6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? => ? = 6
[5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [8,4,9,2,5,10,11,1,3,6,7,12] => ? => ? = 8
[4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> [9,5,10,11,2,6,7,12,1,3,4,8] => ? => ? = 7
[6,5,2]
=> [[1,2,5,6,7,13],[3,4,10,11,12],[8,9]]
=> ? => ? => ? = 8
[6,4,3]
=> [[1,2,3,7,12,13],[4,5,6,11],[8,9,10]]
=> ? => ? => ? = 9
[6,4,2,1]
=> [[1,3,6,7,12,13],[2,5,10,11],[4,9],[8]]
=> ? => ? => ? = 7
[6,2,2,2,1]
=> [[1,3,10,11,12,13],[2,5],[4,7],[6,9],[8]]
=> ? => ? => ? = 9
[6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> ? => ? => ? = 7
[6,5,2,1]
=> [[1,3,6,7,8,14],[2,5,11,12,13],[4,10],[9]]
=> ? => ? => ? = 10
[6,4,3,1]
=> [[1,3,4,8,13,14],[2,6,7,12],[5,10,11],[9]]
=> ? => ? => ? = 8
[6,3,3,2]
=> [[1,2,5,12,13,14],[3,4,8],[6,7,11],[9,10]]
=> ? => ? => ? = 10
[5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ? => ? => ? = 12
[3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? => ? => ? = 10
[6,5,3,1]
=> [[1,3,4,8,9,15],[2,6,7,13,14],[5,11,12],[10]]
=> ? => ? => ? = 7
[5,5,4,1]
=> [[1,3,4,5,10],[2,7,8,9,15],[6,12,13,14],[11]]
=> ? => ? => ? = 10
[4,4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10,15],[12,13,14]]
=> ? => ? => ? = 12
[6,5,3,2]
=> [[1,2,5,9,10,16],[3,4,8,14,15],[6,7,13],[11,12]]
=> ? => ? => ? = 10
[6,5,2,2,1]
=> [[1,3,8,9,10,16],[2,5,13,14,15],[4,7],[6,12],[11]]
=> ? => ? => ? = 11
[5,5,4,2]
=> [[1,2,5,6,11],[3,4,9,10,16],[7,8,14,15],[12,13]]
=> ? => ? => ? = 8
[4,4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11,16],[9,14,15],[13]]
=> ? => ? => ? = 9
[5,5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11,17],[4,9,15,16],[8,14],[13]]
=> ? => ? => ? = 11
Description
The length of the longest increasing subsequence of the permutation.
Matching statistic: St000308
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 3
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [1,3,2] => 2
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 4
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [1,3,4,2] => 3
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,3,4,5,2] => 4
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,2,5,3,4] => 4
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,3,2,5,4] => 3
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,3,4,5,6,2] => 5
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,2,5,6,3,4] => 4
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => ? = 6
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [1,2,5,6,7,3,4] => ? = 5
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => ? = 6
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [1,3,6,7,2,5,4] => ? = 5
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [1,3,2,5,4,7,6] => ? = 4
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [1,2,5,6,7,8,3,4] => ? = 6
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [1,2,3,7,8,4,5,6] => ? = 5
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [1,3,6,7,8,2,5,4] => ? = 6
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [1,3,4,8,2,6,7,5] => ? = 5
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [1,2,5,3,4,8,6,7] => ? = 6
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [1,2,3,7,8,9,4,5,6] => ? = 6
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [1,3,6,7,8,9,2,5,4] => ? = 7
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [1,2,3,4,9,5,6,7,8] => ? = 8
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [1,3,4,8,9,2,6,7,5] => ? = 5
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [1,3,2,5,4,7,6,9,8] => ? = 5
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [1,2,3,4,9,10,5,6,7,8] => ? = 6
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [1,3,4,8,9,10,2,6,7,5] => ? = 6
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5,10] => [1,3,4,5,10,2,7,8,9,6] => ? = 7
[5,3,2]
=> [[1,2,5,9,10],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9,10] => [1,2,5,9,10,3,4,8,6,7] => ? = 7
[5,2,2,1]
=> [[1,3,8,9,10],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9,10] => [1,3,8,9,10,2,5,4,7,6] => ? = 7
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? => ? = 10
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? => ? = 6
[6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? => ? = 8
[6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? => ? => ? = 8
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => ? => ? = 6
[4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? => ? = 9
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? => ? = 8
[2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? => ? = 6
[6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? => ? => ? = 9
[6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? => ? = 6
[5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [8,4,9,2,5,10,11,1,3,6,7,12] => ? => ? = 8
[4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> [9,5,10,11,2,6,7,12,1,3,4,8] => ? => ? = 7
[6,5,2]
=> [[1,2,5,6,7,13],[3,4,10,11,12],[8,9]]
=> ? => ? => ? = 8
[6,4,3]
=> [[1,2,3,7,12,13],[4,5,6,11],[8,9,10]]
=> ? => ? => ? = 9
[6,4,2,1]
=> [[1,3,6,7,12,13],[2,5,10,11],[4,9],[8]]
=> ? => ? => ? = 7
[6,2,2,2,1]
=> [[1,3,10,11,12,13],[2,5],[4,7],[6,9],[8]]
=> ? => ? => ? = 9
[6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> ? => ? => ? = 7
[6,5,2,1]
=> [[1,3,6,7,8,14],[2,5,11,12,13],[4,10],[9]]
=> ? => ? => ? = 10
[6,4,3,1]
=> [[1,3,4,8,13,14],[2,6,7,12],[5,10,11],[9]]
=> ? => ? => ? = 8
[6,3,3,2]
=> [[1,2,5,12,13,14],[3,4,8],[6,7,11],[9,10]]
=> ? => ? => ? = 10
[5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ? => ? => ? = 12
[3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? => ? => ? = 10
[6,5,3,1]
=> [[1,3,4,8,9,15],[2,6,7,13,14],[5,11,12],[10]]
=> ? => ? => ? = 7
[5,5,4,1]
=> [[1,3,4,5,10],[2,7,8,9,15],[6,12,13,14],[11]]
=> ? => ? => ? = 10
[4,4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10,15],[12,13,14]]
=> ? => ? => ? = 12
[6,5,3,2]
=> [[1,2,5,9,10,16],[3,4,8,14,15],[6,7,13],[11,12]]
=> ? => ? => ? = 10
[6,5,2,2,1]
=> [[1,3,8,9,10,16],[2,5,13,14,15],[4,7],[6,12],[11]]
=> ? => ? => ? = 11
[5,5,4,2]
=> [[1,2,5,6,11],[3,4,9,10,16],[7,8,14,15],[12,13]]
=> ? => ? => ? = 8
[4,4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11,16],[9,14,15],[13]]
=> ? => ? => ? = 9
[5,5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11,17],[4,9,15,16],[8,14],[13]]
=> ? => ? => ? = 11
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St001566
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001566: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001566: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 3
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 4
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 3
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 4
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => 4
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,5,4,3,1] => 3
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 5
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => 4
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 6
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [4,3,2,1,5,6,7] => ? = 5
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [6,5,4,3,2,1,7] => ? = 6
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,5,4,3,1,6,7] => ? = 5
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [5,6,4,2,7,3,1] => ? = 4
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [4,3,2,1,5,6,7,8] => ? = 6
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [6,5,4,3,2,1,7,8] => ? = 5
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [2,5,4,3,1,6,7,8] => ? = 6
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [2,7,6,5,4,3,1,8] => ? = 5
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [4,3,7,8,6,5,2,1] => ? = 6
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [6,5,4,3,2,1,7,8,9] => ? = 6
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [2,5,4,3,1,6,7,8,9] => ? = 7
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [8,7,6,5,4,3,2,1,9] => ? = 8
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [2,7,6,5,4,3,1,8,9] => ? = 5
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => [6,7,4,9,8,3,1,5,2] => ? = 5
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9,10] => [8,7,6,5,4,3,2,1,9,10] => ? = 6
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9,10] => [2,7,6,5,4,3,1,8,9,10] => ? = 6
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5,10] => [2,9,8,7,6,5,4,3,1,10] => ? = 7
[5,3,2]
=> [[1,2,5,9,10],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9,10] => [4,3,7,8,6,5,2,1,9,10] => ? = 7
[5,2,2,1]
=> [[1,3,8,9,10],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9,10] => [5,6,4,2,7,3,1,8,9,10] => ? = 7
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? => ? = 10
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? => ? = 6
[6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? => ? = 8
[6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? => ? => ? = 8
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => ? => ? = 6
[4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? => ? = 9
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? => ? = 8
[2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? => ? = 6
[6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? => ? => ? = 9
[6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11,12] => ? => ? = 6
[5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [8,4,9,2,5,10,11,1,3,6,7,12] => ? => ? = 8
[4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> [9,5,10,11,2,6,7,12,1,3,4,8] => ? => ? = 7
[6,5,2]
=> [[1,2,5,6,7,13],[3,4,10,11,12],[8,9]]
=> ? => ? => ? = 8
[6,4,3]
=> [[1,2,3,7,12,13],[4,5,6,11],[8,9,10]]
=> ? => ? => ? = 9
[6,4,2,1]
=> [[1,3,6,7,12,13],[2,5,10,11],[4,9],[8]]
=> ? => ? => ? = 7
[6,2,2,2,1]
=> [[1,3,10,11,12,13],[2,5],[4,7],[6,9],[8]]
=> ? => ? => ? = 9
[6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> ? => ? => ? = 7
[6,5,2,1]
=> [[1,3,6,7,8,14],[2,5,11,12,13],[4,10],[9]]
=> ? => ? => ? = 10
[6,4,3,1]
=> [[1,3,4,8,13,14],[2,6,7,12],[5,10,11],[9]]
=> ? => ? => ? = 8
[6,3,3,2]
=> [[1,2,5,12,13,14],[3,4,8],[6,7,11],[9,10]]
=> ? => ? => ? = 10
[5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ? => ? => ? = 12
[3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? => ? => ? = 10
[6,5,3,1]
=> [[1,3,4,8,9,15],[2,6,7,13,14],[5,11,12],[10]]
=> ? => ? => ? = 7
[5,5,4,1]
=> [[1,3,4,5,10],[2,7,8,9,15],[6,12,13,14],[11]]
=> ? => ? => ? = 10
[4,4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10,15],[12,13,14]]
=> ? => ? => ? = 12
[6,5,3,2]
=> [[1,2,5,9,10,16],[3,4,8,14,15],[6,7,13],[11,12]]
=> ? => ? => ? = 10
[6,5,2,2,1]
=> [[1,3,8,9,10,16],[2,5,13,14,15],[4,7],[6,12],[11]]
=> ? => ? => ? = 11
[5,5,4,2]
=> [[1,2,5,6,11],[3,4,9,10,16],[7,8,14,15],[12,13]]
=> ? => ? => ? = 8
[4,4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11,16],[9,14,15],[13]]
=> ? => ? => ? = 9
[5,5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11,17],[4,9,15,16],[8,14],[13]]
=> ? => ? => ? = 11
Description
The length of the longest arithmetic progression in a permutation.
For a permutation $\pi$ of length $n$, this is the biggest $k$ such that there exist $1 \leq i_1 < \dots < i_k \leq n$ with
$$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$$
Matching statistic: St000923
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000923: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 42%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000923: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 42%
Values
[1]
=> [[1]]
=> [1] => [1] => ? = 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 3
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [3,1,2] => 2
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 4
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [4,1,2,3] => 3
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5,1,2,3,4] => 4
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [5,1,4,2,3] => 4
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,3,1,5,2] => 3
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => 5
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [6,1,5,2,3,4] => 4
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 6
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [7,1,6,2,3,4,5] => ? = 5
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [7,1,6,2,5,3,4] => ? = 6
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [6,1,5,2,7,3,4] => ? = 5
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [6,3,4,5,1,7,2] => ? = 4
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [8,1,7,2,3,4,5,6] => ? = 6
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [8,1,7,2,6,3,4,5] => ? = 5
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [7,1,6,2,8,3,4,5] => ? = 6
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [7,1,6,2,5,3,8,4] => ? = 5
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [6,5,4,1,8,2,7,3] => ? = 6
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [9,1,8,2,7,3,4,5,6] => ? = 6
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [8,1,7,2,9,3,4,5,6] => ? = 7
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [9,1,8,2,7,3,6,4,5] => ? = 8
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [8,1,7,2,6,3,9,4,5] => ? = 5
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => [6,5,3,8,4,7,1,9,2] => ? = 5
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [7,8,9,10,1,2,3,4,5,6] => [10,1,9,2,8,3,7,4,5,6] => ? = 6
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [10,7,8,9,1,2,3,4,5,6] => [9,1,8,2,7,3,10,4,5,6] => ? = 6
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [10,6,7,8,9,1,2,3,4,5] => [9,1,8,2,7,3,6,4,10,5] => ? = 7
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [9,10,6,7,8,1,2,3,4,5] => [8,1,7,2,6,3,10,4,9,5] => ? = 7
[5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,1,2,3,4,5] => [7,1,6,2,9,3,8,4,10,5] => ? = 7
[6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? => ? = 10
[6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? => ? => ? = 6
[6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? => ? => ? = 8
[6,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? => ? => ? = 8
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [10,11,6,7,8,9,1,2,3,4,5] => ? => ? = 6
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? => ? = 9
[3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? => ? = 8
[2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? => ? = 6
[6,5,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12]]
=> ? => ? => ? = 9
[6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [11,12,7,8,9,10,1,2,3,4,5,6] => [10,1,9,2,8,3,7,4,12,5,11,6] => ? = 6
[5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [12,10,11,6,7,8,9,1,2,3,4,5] => ? => ? = 8
[4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [12,9,10,11,5,6,7,8,1,2,3,4] => ? => ? = 7
[6,5,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13]]
=> ? => ? => ? = 8
[6,4,3]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13]]
=> ? => ? => ? = 9
[6,4,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13]]
=> ? => ? => ? = 7
[6,2,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13]]
=> ? => ? => ? = 9
[6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? => ? => ? = 7
[6,5,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13],[14]]
=> ? => ? => ? = 10
[6,4,3,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13],[14]]
=> ? => ? => ? = 8
[6,3,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14]]
=> ? => ? => ? = 10
[5,5,4]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14]]
=> ? => ? => ? = 12
[3,3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14]]
=> ? => ? => ? = 10
[6,5,3,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14],[15]]
=> ? => ? => ? = 7
[5,5,4,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14],[15]]
=> ? => ? => ? = 10
[4,4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15]]
=> ? => ? => ? = 12
[6,5,3,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14],[15,16]]
=> ? => ? => ? = 10
[6,5,2,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13],[14,15],[16]]
=> ? => ? => ? = 11
[5,5,4,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14],[15,16]]
=> ? => ? => ? = 8
[4,4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15],[16]]
=> ? => ? => ? = 9
Description
The minimal number with no two order isomorphic substrings of this length in a permutation.
For example, the length $3$ substrings of the permutation $12435$ are $124$, $243$ and $435$, whereas its length $2$ substrings are $12$, $24$, $43$ and $35$.
No two sequences among $124$, $243$ and $435$ are order isomorphic, but $12$ and $24$ are, so the statistic on $12435$ is $3$.
This is inspired by [[St000922]].
Matching statistic: St001965
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001965: Alternating sign matrices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001965: Alternating sign matrices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 50%
Values
[1]
=> [[1]]
=> [1] => [[1]]
=> 0 = 1 - 1
[2]
=> [[1,2]]
=> [1,2] => [[1,0],[0,1]]
=> 1 = 2 - 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 2 = 3 - 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 1 = 2 - 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 3 = 4 - 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 2 = 3 - 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [[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]]
=> 4 = 5 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 3 = 4 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> 3 = 4 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> 2 = 3 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [[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]]
=> 5 = 6 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[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],[1,0,0,0,0,0]]
=> ? = 5 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> ? = 4 - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [[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],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
=> ? = 6 - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [[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],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0]]
=> ? = 5 - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 6 - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 5 - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 4 - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0]]
=> ? = 6 - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0]]
=> ? = 5 - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0]]
=> ? = 6 - 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0]]
=> ? = 5 - 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0]]
=> ? = 6 - 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0]]
=> ? = 6 - 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0]]
=> ? = 7 - 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0]]
=> ? = 8 - 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0]]
=> ? = 5 - 1
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => [[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0]]
=> ? = 5 - 1
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [7,8,9,10,1,2,3,4,5,6] => [[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0]]
=> ? = 6 - 1
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [10,7,8,9,1,2,3,4,5,6] => [[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0]]
=> ? = 6 - 1
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [10,6,7,8,9,1,2,3,4,5] => [[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0]]
=> ? = 7 - 1
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [9,10,6,7,8,1,2,3,4,5] => [[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0]]
=> ? = 7 - 1
[5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,1,2,3,4,5] => [[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[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],[0,0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0]]
=> ? = 7 - 1
[6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ?
=> ? = 10 - 1
[6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? => ?
=> ? = 6 - 1
[6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? => ?
=> ? = 8 - 1
[6,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? => ?
=> ? = 8 - 1
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [10,11,6,7,8,9,1,2,3,4,5] => ?
=> ? = 6 - 1
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ?
=> ? = 9 - 1
[3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ?
=> ? = 8 - 1
[2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ?
=> ? = 6 - 1
[6,5,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12]]
=> ? => ?
=> ? = 9 - 1
[6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [11,12,7,8,9,10,1,2,3,4,5,6] => [[0,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0]]
=> ? = 6 - 1
[5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [12,10,11,6,7,8,9,1,2,3,4,5] => ?
=> ? = 8 - 1
[4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [12,9,10,11,5,6,7,8,1,2,3,4] => ?
=> ? = 7 - 1
[6,5,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13]]
=> ? => ?
=> ? = 8 - 1
[6,4,3]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13]]
=> ? => ?
=> ? = 9 - 1
[6,4,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13]]
=> ? => ?
=> ? = 7 - 1
[6,2,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13]]
=> ? => ?
=> ? = 9 - 1
[6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? => ?
=> ? = 7 - 1
[6,5,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13],[14]]
=> ? => ?
=> ? = 10 - 1
[6,4,3,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13],[14]]
=> ? => ?
=> ? = 8 - 1
[6,3,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14]]
=> ? => ?
=> ? = 10 - 1
[5,5,4]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14]]
=> ? => ?
=> ? = 12 - 1
[3,3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14]]
=> ? => ?
=> ? = 10 - 1
[6,5,3,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14],[15]]
=> ? => ?
=> ? = 7 - 1
[5,5,4,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14],[15]]
=> ? => ?
=> ? = 10 - 1
[4,4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15]]
=> ? => ?
=> ? = 12 - 1
[6,5,3,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14],[15,16]]
=> ? => ?
=> ? = 10 - 1
[6,5,2,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13],[14,15],[16]]
=> ? => ?
=> ? = 11 - 1
[5,5,4,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14],[15,16]]
=> ? => ?
=> ? = 8 - 1
Description
The number of decreasable positions in the corner sum matrix of an alternating sign matrix.
A decreasable position in a corner sum matrix is an entry such that the matrix obtained by decreasing it by one yields the corner sum matrix of some alternating sign matrix.
Matching statistic: St000327
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> ? = 1 - 1
[2]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> 1 = 2 - 1
[3]
=> [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2 = 3 - 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 3 = 4 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 3 = 4 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 2 = 3 - 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 6 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? = 5 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 4 - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 6 - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 5 - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 6 - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 5 - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 4 - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ? = 6 - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
=> ? = 5 - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
=> ? = 6 - 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ([(1,6),(2,7),(5,4),(6,5),(7,3)],8)
=> ? = 5 - 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ([(0,5),(1,7),(2,6),(6,3),(7,4)],8)
=> ? = 6 - 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => ([(0,8),(1,7),(4,6),(5,4),(6,3),(7,5),(8,2)],9)
=> ? = 6 - 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => ([(1,8),(2,4),(5,7),(6,5),(7,3),(8,6)],9)
=> ? = 7 - 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => ([(0,7),(1,8),(4,5),(5,2),(6,3),(7,6),(8,4)],9)
=> ? = 8 - 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => ([(1,8),(2,7),(5,6),(6,4),(7,5),(8,3)],9)
=> ? = 5 - 1
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => ([(1,8),(2,7),(3,6),(4,5)],9)
=> ? = 5 - 1
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [7,8,9,10,1,2,3,4,5,6] => ([(0,8),(1,9),(4,6),(5,4),(6,3),(7,2),(8,7),(9,5)],10)
=> ? = 6 - 1
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [10,7,8,9,1,2,3,4,5,6] => ([(1,9),(2,8),(5,7),(6,5),(7,4),(8,6),(9,3)],10)
=> ? = 6 - 1
[5,4,1]
=> [[1,2,3,4,5],[6,7,8,9],[10]]
=> [10,6,7,8,9,1,2,3,4,5] => ([(1,8),(2,9),(5,6),(6,3),(7,4),(8,7),(9,5)],10)
=> ? = 7 - 1
[5,3,2]
=> [[1,2,3,4,5],[6,7,8],[9,10]]
=> [9,10,6,7,8,1,2,3,4,5] => ([(0,9),(1,8),(2,5),(6,7),(7,4),(8,6),(9,3)],10)
=> ? = 7 - 1
[5,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10]]
=> [10,8,9,6,7,1,2,3,4,5] => ([(1,6),(2,5),(3,9),(7,8),(8,4),(9,7)],10)
=> ? = 7 - 1
[6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ?
=> ? = 10 - 1
[6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? => ?
=> ? = 6 - 1
[6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? => ?
=> ? = 8 - 1
[6,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? => ?
=> ? = 8 - 1
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [10,11,6,7,8,9,1,2,3,4,5] => ?
=> ? = 6 - 1
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ?
=> ? = 9 - 1
[3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ?
=> ? = 8 - 1
[2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ?
=> ? = 6 - 1
[6,5,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12]]
=> ? => ?
=> ? = 9 - 1
[6,4,2]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12]]
=> [11,12,7,8,9,10,1,2,3,4,5,6] => ([(0,10),(1,11),(2,5),(6,8),(7,6),(8,4),(9,3),(10,9),(11,7)],12)
=> ? = 6 - 1
[5,4,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12]]
=> [12,10,11,6,7,8,9,1,2,3,4,5] => ?
=> ? = 8 - 1
[4,4,3,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12]]
=> [12,9,10,11,5,6,7,8,1,2,3,4] => ?
=> ? = 7 - 1
[6,5,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13]]
=> ? => ?
=> ? = 8 - 1
[6,4,3]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13]]
=> ? => ?
=> ? = 9 - 1
[6,4,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12],[13]]
=> ? => ?
=> ? = 7 - 1
[6,2,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13]]
=> ? => ?
=> ? = 9 - 1
[6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? => ?
=> ? = 7 - 1
[6,5,2,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13],[14]]
=> ? => ?
=> ? = 10 - 1
[6,4,3,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13],[14]]
=> ? => ?
=> ? = 8 - 1
[6,3,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14]]
=> ? => ?
=> ? = 10 - 1
[5,5,4]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14]]
=> ? => ?
=> ? = 12 - 1
[3,3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14]]
=> ? => ?
=> ? = 10 - 1
[6,5,3,1]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14],[15]]
=> ? => ?
=> ? = 7 - 1
[5,5,4,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14],[15]]
=> ? => ?
=> ? = 10 - 1
[4,4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15]]
=> ? => ?
=> ? = 12 - 1
[6,5,3,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14],[15,16]]
=> ? => ?
=> ? = 10 - 1
Description
The number of cover relations in a poset.
Equivalently, this is also the number of edges in the Hasse diagram [1].
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001668The number of points of the poset minus the width of the poset. 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. St001637The number of (upper) dissectors of a poset. St001626The number of maximal proper sublattices of a lattice. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000214The number of adjacencies of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000742The number of big ascents of a permutation after prepending zero. St001096The size of the overlap set of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000942The number of critical left to right maxima of the parking functions. St001645The pebbling number of a connected graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001937The size of the center of a parking function. St000173The segment statistic of a semistandard tableau. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001935The number of ascents in a parking function.
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