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Your data matches 106 different statistics following compositions of up to 3 maps.
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Matching statistic: St001880
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[[[.,.],[.,[[.,.],.]]],.]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 4
[[[.,.],[[.,.],[.,.]]],.]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[[[.,.],[[.,[.,.]],.]],.]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
[[[.,.],[[[.,.],.],.]],.]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[[[[.,.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[[[[.,.],.],[[.,.],.]],.]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[[[[[.,.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
[[[[.,.],[.,[.,.]]],.],.]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[[[[.,.],[[.,.],.]],.],.]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[[[[[.,.],.],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[[[[[.,.],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[[[[[[.,.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 4
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [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
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [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
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,4,5,6,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 4
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [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
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [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
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [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
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [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
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [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
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [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
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 4
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 5
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> 5
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [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
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [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
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 5
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> 6
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [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
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 7
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> 5
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> 5
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [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
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [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
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> 5
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 6
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 7
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000907
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%
Values
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 4
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? = 4
[[[.,.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 4
[[[.,.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 6
[[[.,.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 6
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 4
[[[[.,.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? = 5
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[[[.,.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5
[[[[.,.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 5
[[[[[.,.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[[[[.,.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[[[[[.,.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 4
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> ? = 4
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> ? = 6
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 4
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [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
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> ? = 6
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> ? = 6
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 6
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> ? = 4
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> ? = 5
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 5
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 7
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 7
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> ? = 5
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 6
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> ? = 6
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> ? = 5
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 5
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 7
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 7
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ? = 5
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 6
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> ? = 6
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? = 6
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St001879
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 100%
Values
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 - 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 - 1
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 4 - 1
[[[.,.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 4 - 1
[[[.,.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [[.,[[.,.],[[.,.],.]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 6 - 1
[[[.,.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? = 6 - 1
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[.,[[.,.],[.,[.,.]]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4 - 1
[[[[.,.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 5 - 1
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [[.,[[[.,.],.],[.,.]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 5 - 1
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [[.,[[[.,[.,.]],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[[[[.,.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 5 - 1
[[[[.,.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 5 - 1
[[[[[.,.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[[[[[.,.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[[[[[[.,.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[.,[[.,.],.]],[.,[.,[.,.]]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ? = 4 - 1
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ? = 4 - 1
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],.]],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ? = 6 - 1
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [[.,[[.,.],.]],[[.,.],[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7)
=> ? = 6 - 1
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [[.,[[.,.],.]],[[.,[.,.]],.]]
=> ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
=> ? = 4 - 1
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[[.,.],.]]],[.,.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 6 - 1
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [[.,[[.,.],[[.,.],[.,.]]]],.]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ? = 6 - 1
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [[.,[[.,.],[[[.,.],.],.]]],.]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 6 - 1
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [[.,[[.,.],[.,.]]],[.,[.,.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> ? = 6 - 1
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],[.,.]]],[[.,.],.]]
=> ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7)
=> ? = 6 - 1
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [[.,[[.,.],[.,[.,.]]]],[.,.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 6 - 1
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 4 - 1
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [[.,[[[.,.],.],.]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 5 - 1
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [[.,[[[.,.],.],.]],[[.,.],.]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 5 - 1
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [[.,[[[.,.],.],[[.,.],.]]],.]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 7 - 1
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [[.,[[[.,.],.],[.,.]]],[.,.]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 7 - 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [[.,[[[.,.],.],[.,[.,.]]]],.]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 5 - 1
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [[.,[[[.,[.,.]],.],.]],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 6 - 1
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [[.,[[[.,[.,.]],.],[.,.]]],.]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 6 - 1
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [[.,[[[.,[[.,.],.]],.],.]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [[.,[[[.,[.,[.,.]]],.],.]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[.,[.,[[.,.],.]]],[.,[.,.]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 5 - 1
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],[[.,.],.]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 5 - 1
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [[.,[.,[[.,.],[[.,.],.]]]],.]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 7 - 1
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [[.,[.,[[.,.],[.,.]]]],[.,.]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ? = 7 - 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[[[.,.],.],.]]],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 6 - 1
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [[.,[.,[[[.,.],.],[.,.]]]],.]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 6 - 1
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [[.,[.,[[[.,[.,.]],.],.]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[.,[.,[.,[[.,.],.]]]],[.,.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 6 - 1
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 6 - 1
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [[.,[.,[.,[[[.,.],.],.]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St000725
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000725: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 4
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 4
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 4
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 5
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 5
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 5
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? = 4
[[[.,.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 4
[[[.,.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 6
[[[.,.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 6
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ? = 4
[[[[.,.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 5
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 5
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 6
[[[[.,.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? = 5
[[[[.,.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => ? = 5
[[[[[.,.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => 6
[[[[[.,.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => 6
[[[[[[.,.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => ? = 6
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [6,3,1,5,2,7,8,4] => ? = 4
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [4,3,1,6,2,7,8,5] => ? = 4
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [4,3,1,7,2,5,8,6] => ? = 6
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [8,3,1,6,2,7,4,5] => ? = 6
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,1,8,2,7,5,6] => ? = 4
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [8,3,1,5,2,7,4,6] => ? = 6
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,1,5,2,8,4,7] => ? = 6
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [4,3,1,8,2,5,6,7] => ? = 6
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [7,3,1,6,2,4,8,5] => ? = 6
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [7,3,1,5,2,4,8,6] => ? = 6
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [7,3,1,8,2,4,5,6] => ? = 6
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [8,3,1,6,2,4,5,7] => ? = 4
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,1,7,6,2,8,5] => ? = 5
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,1,5,7,2,8,6] => ? = 5
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,1,5,8,2,6,7] => ? = 7
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,1,8,7,2,5,6] => ? = 7
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,1,8,6,2,5,7] => ? = 5
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [8,4,1,5,7,2,3,6] => ? = 6
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [8,4,1,5,6,2,3,7] => ? = 6
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ? = 7
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,8,1,3,6,4,5,7] => ? = 7
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [7,4,1,2,6,3,8,5] => ? = 5
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [5,4,1,2,7,3,8,6] => ? = 5
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,8,3,6,7] => ? = 7
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [8,4,1,2,7,3,5,6] => ? = 7
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [8,4,1,2,6,3,5,7] => ? = 5
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [5,4,1,2,8,7,3,6] => ? = 6
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [5,4,1,2,6,8,3,7] => ? = 6
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,3,1,2,4,8,5,7] => ? = 7
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => ? = 6
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => ? = 6
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => 7
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [5,8,1,2,3,4,6,7] => ? = 7
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => ? = 7
Description
The smallest label of a leaf of the increasing binary tree associated to a permutation.
Matching statistic: St000308
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 80%
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 80%
Values
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 3 = 4 - 1
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3 = 4 - 1
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 3 = 4 - 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 3 = 4 - 1
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 4 = 5 - 1
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 4 = 5 - 1
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 4 = 5 - 1
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 4 - 1
[[[.,.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => ? = 4 - 1
[[[.,.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 6 - 1
[[[.,.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 6 - 1
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 4 - 1
[[[[.,.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 5 - 1
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 5 - 1
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 6 - 1
[[[[.,.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 5 - 1
[[[[.,.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 5 - 1
[[[[[.,.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => 5 = 6 - 1
[[[[[.,.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => 5 = 6 - 1
[[[[[[.,.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 5 = 6 - 1
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,4,8,2,5,6,7] => ? = 4 - 1
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [3,1,4,8,7,2,5,6] => ? = 4 - 1
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => ? = 6 - 1
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [3,1,4,8,6,2,5,7] => ? = 6 - 1
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,1,4,8,2,7,5,6] => ? = 4 - 1
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,5,2,6,8,4,7] => ? = 6 - 1
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => ? = 6 - 1
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ? = 6 - 1
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,5,2,8,4,6,7] => ? = 6 - 1
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [3,1,5,2,8,7,4,6] => ? = 6 - 1
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => ? = 6 - 1
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,1,7,2,4,5,8,6] => ? = 4 - 1
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [3,1,4,5,8,2,6,7] => ? = 5 - 1
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => ? = 5 - 1
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,4,6,2,7,8,5] => ? = 7 - 1
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,1,4,6,2,8,5,7] => ? = 7 - 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ? = 5 - 1
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,1,4,5,2,8,3,7] => ? = 6 - 1
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [7,1,4,5,2,3,8,6] => ? = 6 - 1
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => ? = 7 - 1
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => ? = 7 - 1
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ? = 5 - 1
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,8,7,3,6] => ? = 5 - 1
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,1,2,6,3,7,8,5] => ? = 7 - 1
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => ? = 7 - 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ? = 5 - 1
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => ? = 6 - 1
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,1,2,5,7,3,8,6] => ? = 6 - 1
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [7,1,2,5,6,3,8,4] => ? = 7 - 1
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,1,2,3,6,8,4,7] => ? = 6 - 1
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,1,2,3,7,4,8,6] => ? = 6 - 1
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [5,1,2,3,6,7,8,4] => ? = 7 - 1
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,1,2,3,4,7,8,5] => ? = 7 - 1
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => ? = 7 - 1
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: St000374
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 3 = 4 - 1
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 3 = 4 - 1
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 3 = 4 - 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 3 = 4 - 1
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 4 = 5 - 1
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 4 = 5 - 1
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 4 = 5 - 1
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? = 4 - 1
[[[.,.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 4 - 1
[[[.,.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 6 - 1
[[[.,.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 6 - 1
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ? = 4 - 1
[[[[.,.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 5 - 1
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 5 - 1
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 6 - 1
[[[[.,.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? = 5 - 1
[[[[.,.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => ? = 5 - 1
[[[[[.,.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => 5 = 6 - 1
[[[[[.,.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => 5 = 6 - 1
[[[[[[.,.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => ? = 6 - 1
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [6,3,1,5,2,7,8,4] => ? = 4 - 1
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [4,3,1,6,2,7,8,5] => ? = 4 - 1
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [4,3,1,7,2,5,8,6] => ? = 6 - 1
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [8,3,1,6,2,7,4,5] => ? = 6 - 1
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,1,8,2,7,5,6] => ? = 4 - 1
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [8,3,1,5,2,7,4,6] => ? = 6 - 1
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,1,5,2,8,4,7] => ? = 6 - 1
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [4,3,1,8,2,5,6,7] => ? = 6 - 1
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [7,3,1,6,2,4,8,5] => ? = 6 - 1
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [7,3,1,5,2,4,8,6] => ? = 6 - 1
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [7,3,1,8,2,4,5,6] => ? = 6 - 1
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [8,3,1,6,2,4,5,7] => ? = 4 - 1
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,1,7,6,2,8,5] => ? = 5 - 1
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,1,5,7,2,8,6] => ? = 5 - 1
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,1,5,8,2,6,7] => ? = 7 - 1
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,1,8,7,2,5,6] => ? = 7 - 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,1,8,6,2,5,7] => ? = 5 - 1
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [8,4,1,5,7,2,3,6] => ? = 6 - 1
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [8,4,1,5,6,2,3,7] => ? = 6 - 1
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ? = 7 - 1
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,8,1,3,6,4,5,7] => ? = 7 - 1
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [7,4,1,2,6,3,8,5] => ? = 5 - 1
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [5,4,1,2,7,3,8,6] => ? = 5 - 1
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,8,3,6,7] => ? = 7 - 1
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [8,4,1,2,7,3,5,6] => ? = 7 - 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [8,4,1,2,6,3,5,7] => ? = 5 - 1
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [5,4,1,2,8,7,3,6] => ? = 6 - 1
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [5,4,1,2,6,8,3,7] => ? = 6 - 1
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,3,1,2,4,8,5,7] => ? = 7 - 1
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => ? = 6 - 1
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => ? = 6 - 1
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => 6 = 7 - 1
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [5,8,1,2,3,4,6,7] => ? = 7 - 1
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => ? = 7 - 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000703
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 3 = 4 - 1
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 3 = 4 - 1
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 3 = 4 - 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 3 = 4 - 1
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 4 = 5 - 1
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 4 = 5 - 1
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 4 = 5 - 1
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? = 4 - 1
[[[.,.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => ? = 4 - 1
[[[.,.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 6 - 1
[[[.,.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [7,3,1,6,2,4,5] => ? = 6 - 1
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => ? = 4 - 1
[[[[.,.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 5 - 1
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 5 - 1
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 6 - 1
[[[[.,.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => ? = 5 - 1
[[[[.,.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => ? = 5 - 1
[[[[[.,.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => 5 = 6 - 1
[[[[[.,.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => 5 = 6 - 1
[[[[[[.,.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => ? = 6 - 1
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [6,3,1,5,2,7,8,4] => ? = 4 - 1
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [4,3,1,6,2,7,8,5] => ? = 4 - 1
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [4,3,1,7,2,5,8,6] => ? = 6 - 1
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [8,3,1,6,2,7,4,5] => ? = 6 - 1
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,1,8,2,7,5,6] => ? = 4 - 1
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [8,3,1,5,2,7,4,6] => ? = 6 - 1
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,1,5,2,8,4,7] => ? = 6 - 1
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [4,3,1,8,2,5,6,7] => ? = 6 - 1
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [7,3,1,6,2,4,8,5] => ? = 6 - 1
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [7,3,1,5,2,4,8,6] => ? = 6 - 1
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [7,3,1,8,2,4,5,6] => ? = 6 - 1
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [8,3,1,6,2,4,5,7] => ? = 4 - 1
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,1,7,6,2,8,5] => ? = 5 - 1
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,1,5,7,2,8,6] => ? = 5 - 1
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,1,5,8,2,6,7] => ? = 7 - 1
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,1,8,7,2,5,6] => ? = 7 - 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,1,8,6,2,5,7] => ? = 5 - 1
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [8,4,1,5,7,2,3,6] => ? = 6 - 1
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [8,4,1,5,6,2,3,7] => ? = 6 - 1
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ? = 7 - 1
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,8,1,3,6,4,5,7] => ? = 7 - 1
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [7,4,1,2,6,3,8,5] => ? = 5 - 1
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [5,4,1,2,7,3,8,6] => ? = 5 - 1
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,8,3,6,7] => ? = 7 - 1
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [8,4,1,2,7,3,5,6] => ? = 7 - 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [8,4,1,2,6,3,5,7] => ? = 5 - 1
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [5,4,1,2,8,7,3,6] => ? = 6 - 1
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [5,4,1,2,6,8,3,7] => ? = 6 - 1
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,3,1,2,4,8,5,7] => ? = 7 - 1
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => ? = 6 - 1
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => ? = 6 - 1
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => 6 = 7 - 1
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [5,8,1,2,3,4,6,7] => ? = 7 - 1
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => ? = 7 - 1
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St001090
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001090: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001090: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 3 = 4 - 1
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 3 = 4 - 1
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 3 = 4 - 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 3 = 4 - 1
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 4 = 5 - 1
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 4 = 5 - 1
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 4 = 5 - 1
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => ? = 4 - 1
[[[.,.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => ? = 4 - 1
[[[.,.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => ? = 6 - 1
[[[.,.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => ? = 6 - 1
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => ? = 4 - 1
[[[[.,.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 5 - 1
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => ? = 5 - 1
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => ? = 6 - 1
[[[[.,.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => ? = 5 - 1
[[[[.,.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => ? = 5 - 1
[[[[[.,.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => ? = 6 - 1
[[[[[.,.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => 5 = 6 - 1
[[[[[[.,.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => 5 = 6 - 1
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [4,3,1,8,2,5,6,7] => ? = 4 - 1
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [8,3,4,1,7,2,5,6] => ? = 4 - 1
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [8,3,4,6,1,7,2,5] => ? = 6 - 1
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [8,3,4,1,6,2,5,7] => ? = 6 - 1
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [4,3,1,8,2,7,5,6] => ? = 4 - 1
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,1,5,2,8,4,7] => ? = 6 - 1
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [7,3,1,5,2,4,8,6] => ? = 6 - 1
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [6,3,1,5,2,7,8,4] => ? = 6 - 1
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [5,3,1,2,8,4,6,7] => ? = 6 - 1
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [8,3,1,5,2,7,4,6] => ? = 6 - 1
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [6,3,1,2,4,8,5,7] => ? = 6 - 1
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [7,3,1,2,4,5,8,6] => ? = 4 - 1
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,1,5,8,2,6,7] => ? = 5 - 1
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [5,3,4,1,8,7,2,6] => ? = 5 - 1
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [4,3,1,7,6,2,8,5] => ? = 7 - 1
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [6,3,4,1,2,8,5,7] => ? = 7 - 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,1,0,0]
=> [7,3,4,1,2,5,8,6] => ? = 5 - 1
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,1,4,5,2,8,3,7] => ? = 6 - 1
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,1,4,2,7,3,8,6] => ? = 6 - 1
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [7,4,1,5,6,2,8,3] => ? = 7 - 1
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [6,1,2,5,3,7,8,4] => ? = 7 - 1
[[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,8,3,6,7] => ? = 5 - 1
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [8,5,4,1,2,7,3,6] => ? = 5 - 1
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [7,4,1,2,6,3,8,5] => ? = 7 - 1
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,1,2,3,8,5,7] => ? = 7 - 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [7,4,1,2,3,5,8,6] => ? = 5 - 1
[[[[[.,.],.],[.,[.,.]]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [5,4,1,2,6,8,3,7] => ? = 6 - 1
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [7,5,4,1,2,3,8,6] => ? = 6 - 1
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [7,1,6,5,2,3,8,4] => ? = 7 - 1
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => ? = 6 - 1
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [5,7,1,2,3,4,8,6] => ? = 6 - 1
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,1,2,3,7,8,4] => ? = 7 - 1
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [7,6,1,2,3,4,8,5] => ? = 7 - 1
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,1,2,3,4,5,8,6] => 6 = 7 - 1
Description
The number of pop-stack-sorts needed to sort a permutation.
The pop-stack sorting operator is defined as follows. Process the permutation $\pi$ from left to right. If the stack is empty or its top element is smaller than the current element, empty the stack completely and append its elements to the output in reverse order. Next, push the current element onto the stack. After having processed the last entry, append the stack to the output in reverse order.
A permutation is $t$-pop-stack sortable if it is sortable using $t$ pop-stacks in series.
Matching statistic: St000672
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Values
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 3 - 1
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 3 = 4 - 1
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 3 = 4 - 1
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 3 = 4 - 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 3 = 4 - 1
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 4 = 5 - 1
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 4 = 5 - 1
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 4 = 5 - 1
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 - 1
[[[.,.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? = 4 - 1
[[[.,.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => ? = 6 - 1
[[[.,.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 6 - 1
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 4 - 1
[[[[.,.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 5 - 1
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => ? = 5 - 1
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 6 - 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]
=> [3,1,4,7,6,2,5] => ? = 5 - 1
[[[[.,.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 5 - 1
[[[[[.,.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 6 - 1
[[[[[.,.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 6 - 1
[[[[[[.,.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 5 = 6 - 1
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => ? = 4 - 1
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [8,4,1,6,2,7,3,5] => ? = 4 - 1
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [5,8,1,2,7,3,4,6] => ? = 6 - 1
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [8,4,1,7,6,2,3,5] => ? = 6 - 1
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [8,3,5,1,7,2,4,6] => ? = 4 - 1
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [5,4,1,2,8,7,3,6] => ? = 6 - 1
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,5,4,1,2,8,3,7] => ? = 6 - 1
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,1,5,2,3,8,4,7] => ? = 6 - 1
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [4,3,1,8,6,7,2,5] => ? = 6 - 1
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [8,5,4,1,7,2,3,6] => ? = 6 - 1
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [5,3,4,1,8,7,2,6] => ? = 6 - 1
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => ? = 4 - 1
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,1,2,8,6,7,3,5] => ? = 5 - 1
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [8,1,5,2,7,3,4,6] => ? = 5 - 1
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => ? = 7 - 1
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,1,4,2,8,7,3,6] => ? = 7 - 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,1,4,5,2,8,3,7] => ? = 5 - 1
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,5,1,3,8,7,4,6] => ? = 6 - 1
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,6,1,5,3,8,4,7] => ? = 6 - 1
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => ? = 7 - 1
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? = 7 - 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]
=> [3,1,4,8,6,7,2,5] => ? = 5 - 1
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [8,4,1,5,7,2,3,6] => ? = 5 - 1
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [5,4,1,2,6,8,3,7] => ? = 7 - 1
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [4,3,1,5,8,7,2,6] => ? = 7 - 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => ? = 5 - 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]
=> [4,1,2,5,8,7,3,6] => ? = 6 - 1
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,4,2,6,8,3,7] => ? = 6 - 1
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ? = 7 - 1
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => ? = 6 - 1
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [4,3,1,5,6,8,2,7] => ? = 6 - 1
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => ? = 7 - 1
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,4,5,6,8,2,7] => ? = 7 - 1
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => 6 = 7 - 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St001004
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Values
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 4 = 3 + 1
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 5 = 4 + 1
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 5 = 4 + 1
[[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 5 = 4 + 1
[[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 5 = 4 + 1
[[[[.,.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 6 = 5 + 1
[[[[.,.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 6 = 5 + 1
[[[[[.,.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => 6 = 5 + 1
[[[.,.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => ? = 4 + 1
[[[.,.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [7,4,1,6,2,3,5] => ? = 4 + 1
[[[.,.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => ? = 6 + 1
[[[.,.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => ? = 6 + 1
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => ? = 4 + 1
[[[[.,.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 5 + 1
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => ? = 5 + 1
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 6 + 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]
=> [3,1,4,7,6,2,5] => ? = 5 + 1
[[[[.,.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => ? = 5 + 1
[[[[[.,.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 6 + 1
[[[[[.,.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => ? = 6 + 1
[[[[[[.,.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => 7 = 6 + 1
[[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => ? = 4 + 1
[[[.,.],[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [8,4,1,6,2,7,3,5] => ? = 4 + 1
[[[.,.],[.,[[.,.],[.,.]]]],.]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [5,8,1,2,7,3,4,6] => ? = 6 + 1
[[[.,.],[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [8,4,1,7,6,2,3,5] => ? = 6 + 1
[[[.,.],[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [8,3,5,1,7,2,4,6] => ? = 4 + 1
[[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [5,4,1,2,8,7,3,6] => ? = 6 + 1
[[[.,.],[[.,.],[[.,.],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,5,4,1,2,8,3,7] => ? = 6 + 1
[[[.,.],[[[.,.],.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [6,1,5,2,3,8,4,7] => ? = 6 + 1
[[[.,.],[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [4,3,1,8,6,7,2,5] => ? = 6 + 1
[[[.,.],[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [8,5,4,1,7,2,3,6] => ? = 6 + 1
[[[.,.],[[[.,[.,.]],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [5,3,4,1,8,7,2,6] => ? = 6 + 1
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => ? = 4 + 1
[[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,1,2,8,6,7,3,5] => ? = 5 + 1
[[[[.,.],.],[.,[[.,.],.]]],.]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [8,1,5,2,7,3,4,6] => ? = 5 + 1
[[[[.,.],.],[[.,.],[.,.]]],.]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => ? = 7 + 1
[[[[.,.],.],[[.,[.,.]],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [5,1,4,2,8,7,3,6] => ? = 7 + 1
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,1,4,5,2,8,3,7] => ? = 5 + 1
[[[[[.,.],.],.],[.,[.,.]]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,5,1,3,8,7,4,6] => ? = 6 + 1
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,6,1,5,3,8,4,7] => ? = 6 + 1
[[[[[.,.],[.,.]],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,6,2,4,8,5,7] => ? = 7 + 1
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? = 7 + 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]
=> [3,1,4,8,6,7,2,5] => ? = 5 + 1
[[[[.,.],[.,[[.,.],.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [8,4,1,5,7,2,3,6] => ? = 5 + 1
[[[[.,.],[[.,.],[.,.]]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [5,4,1,2,6,8,3,7] => ? = 7 + 1
[[[[.,.],[[.,[.,.]],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [4,3,1,5,8,7,2,6] => ? = 7 + 1
[[[[.,.],[[[.,.],.],.]],.],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => ? = 5 + 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]
=> [4,1,2,5,8,7,3,6] => ? = 6 + 1
[[[[[.,.],.],[[.,.],.]],.],.]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1,4,2,6,8,3,7] => ? = 6 + 1
[[[[[[.,.],.],.],[.,.]],.],.]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ? = 7 + 1
[[[[[.,.],[.,[.,.]]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => ? = 6 + 1
[[[[[.,.],[[.,.],.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [4,3,1,5,6,8,2,7] => ? = 6 + 1
[[[[[[.,.],.],[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,5,6,8,3,7] => ? = 7 + 1
[[[[[[.,.],[.,.]],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [3,1,4,5,6,8,2,7] => ? = 7 + 1
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => 8 = 7 + 1
Description
The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].
The following 96 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000242The number of indices that are not cyclical small weak excedances. St001180Number of indecomposable injective modules with projective dimension at most 1. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000144The pyramid weight of the Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000653The last descent of a permutation. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000863The length of the first row of the shifted shape of a permutation. St000906The length of the shortest maximal chain in a poset. 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. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001497The position of the largest weak excedence 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. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001655The general position number of a graph. St000015The number of peaks of a Dyck path. St000039The number of crossings of a permutation. St000050The depth or height of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000331The number of upper interactions of a Dyck path. St000643The size of the largest orbit of antichains under Panyushev complementation. St000673The number of non-fixed points of a permutation. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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. St001566The length of the longest arithmetic progression in a permutation. St001668The number of points of the poset minus the width of the poset. St001948The number of augmented double ascents of a permutation. St000314The number of left-to-right-maxima of a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000365The number of double ascents of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001405The number of bonds in a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000080The rank of the poset. St000454The largest eigenvalue of a graph if it is integral. 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 $eA$. St001812The biclique partition number of a graph. St001875The number of simple modules with projective dimension at most 1. St001623The number of doubly irreducible elements of a lattice. St000019The cardinality of the support of a permutation. St000327The number of cover relations in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001637The number of (upper) dissectors of a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000201The number of leaf nodes in a binary tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000742The number of big ascents of a permutation after prepending zero. St000911The number of maximal antichains of maximal size in a poset. St000996The number of exclusive left-to-right maxima of a permutation. St001330The hat guessing number of a graph. St000245The number of ascents of a permutation. St000356The number of occurrences of the pattern 13-2. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000834The number of right outer peaks of a permutation. St001782The order of rowmotion on the set of order ideals of a poset.
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