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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St001066
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St000118
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [.,.]
=> 0 = 1 - 1
[1,2] => [.,[.,.]]
=> [.,[.,.]]
=> 0 = 1 - 1
[2,1] => [[.,.],.]
=> [[.,.],.]
=> 0 = 1 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> 1 = 2 - 1
[1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> 0 = 1 - 1
[2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0 = 1 - 1
[2,3,1] => [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> 0 = 1 - 1
[3,1,2] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0 = 1 - 1
[3,2,1] => [[[.,.],.],.]
=> [[[.,.],.],.]
=> 0 = 1 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> 2 = 3 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> 1 = 2 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1 = 2 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0 = 1 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> 0 = 1 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1 = 2 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0 = 1 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0 = 1 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> 0 = 1 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 0 = 1 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> 0 = 1 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1 = 2 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0 = 1 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0 = 1 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> 0 = 1 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0 = 1 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> 0 = 1 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> 3 = 4 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> 2 = 3 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> 1 = 2 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> 2 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> 1 = 2 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> 1 = 2 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 1 = 2 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 0 = 1 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> 1 = 2 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> 2 = 3 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> 1 = 2 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> 1 = 2 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 1 = 2 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 0 = 1 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> 0 = 1 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> 0 = 1 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> 1 = 2 - 1
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree.
[[oeis:A001006]] counts binary trees avoiding this pattern.
Matching statistic: St001483
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1,0]
=> 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St001167
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 0 = 1 - 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001253
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 0 = 1 - 1
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details.
The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
Matching statistic: St000931
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> ? = 1 - 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 0 = 1 - 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0 = 1 - 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3 = 4 - 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0 = 1 - 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
Description
The number of occurrences of the pattern UUU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St000366
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,2] => [2] => [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1 = 2 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 3 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 2 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 2 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 2 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 3 = 4 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 2 = 3 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 2 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 2 = 3 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 2 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1 = 2 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 2 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 2 = 3 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 2 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1 = 2 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 2 - 1
[1,2,3,5,4,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,3,5,4,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,3,6,4,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,3,6,4,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,3,6,5,4,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,3,6,5,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,3,7,4,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,3,7,4,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,3,7,5,4,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,3,7,5,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,3,7,6,4,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,4,3,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 4 - 1
[1,2,4,3,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[1,2,4,3,6,5,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,4,3,6,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[1,2,4,3,7,5,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,4,3,7,6,5] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 2 - 1
[1,2,4,5,3,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,4,5,3,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,4,6,3,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,4,6,3,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,4,6,5,3,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,4,6,5,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,4,7,3,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,4,7,3,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,4,7,5,3,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,4,7,5,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,4,7,6,3,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,5,3,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 4 - 1
[1,2,5,3,4,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[1,2,5,3,6,4,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,5,3,6,7,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[1,2,5,3,7,4,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,5,3,7,6,4] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 2 - 1
[1,2,5,4,3,6,7] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 3 - 1
[1,2,5,4,3,7,6] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 2 - 1
[1,2,5,4,6,3,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,5,4,6,7,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[1,2,5,4,7,3,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,5,4,7,6,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 2 - 1
[1,2,5,6,3,4,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,5,6,3,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,5,6,4,3,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,5,6,4,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,5,7,3,4,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,5,7,3,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,5,7,4,3,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,5,7,4,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,5,7,6,3,4] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,6,3,4,5,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 4 - 1
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St000371
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,2] => [2] => [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1 = 2 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 3 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 2 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 2 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 2 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1 = 2 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 3 = 4 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 2 = 3 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 2 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 2 = 3 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 2 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1 = 2 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 2 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 2 = 3 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 2 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1 = 2 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1 = 2 - 1
[1,2,3,5,4,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,3,5,4,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,3,6,4,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,3,6,4,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,3,6,5,4,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,3,6,5,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,3,7,4,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,3,7,4,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,3,7,5,4,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,3,7,5,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,3,7,6,4,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,4,3,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 4 - 1
[1,2,4,3,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[1,2,4,3,6,5,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,4,3,6,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[1,2,4,3,7,5,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,4,3,7,6,5] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 2 - 1
[1,2,4,5,3,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,4,5,3,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,4,6,3,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,4,6,3,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,4,6,5,3,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,4,6,5,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,4,7,3,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,4,7,3,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,4,7,5,3,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,4,7,5,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,4,7,6,3,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,5,3,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 4 - 1
[1,2,5,3,4,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[1,2,5,3,6,4,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,5,3,6,7,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[1,2,5,3,7,4,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,5,3,7,6,4] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 2 - 1
[1,2,5,4,3,6,7] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 3 - 1
[1,2,5,4,3,7,6] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 2 - 1
[1,2,5,4,6,3,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,5,4,6,7,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 3 - 1
[1,2,5,4,7,3,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 2 - 1
[1,2,5,4,7,6,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 2 - 1
[1,2,5,6,3,4,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,5,6,3,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,5,6,4,3,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,5,6,4,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,5,7,3,4,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 4 - 1
[1,2,5,7,3,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,5,7,4,3,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,5,7,4,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 3 - 1
[1,2,5,7,6,3,4] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 3 - 1
[1,2,6,3,4,5,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 4 - 1
Description
The number of mid points of decreasing subsequences of length 3 in a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the number of indices $j$ such that there exist indices $i,k$ with $i < j < k$ and $\pi(i) > \pi(j) > \pi(k)$. In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima.
This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence.
See also [[St000119]].
Matching statistic: St000358
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 83%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,2] => [2] => [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 1 = 2 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 2 = 3 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 3 = 4 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2 = 3 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2 = 3 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2 = 3 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,2,3,4,5,6,7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => ? = 6 - 1
[1,2,3,4,5,7,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 5 - 1
[1,2,3,4,6,5,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,3,4,6,7,5] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 5 - 1
[1,2,3,4,7,5,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,3,4,7,6,5] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 4 - 1
[1,2,3,5,4,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,3,5,4,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,3,5,6,4,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,3,5,6,7,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 5 - 1
[1,2,3,5,7,4,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,3,5,7,6,4] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 4 - 1
[1,2,3,6,4,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,3,6,4,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,3,6,5,4,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,3,6,5,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,3,6,7,4,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,3,6,7,5,4] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 4 - 1
[1,2,3,7,4,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,3,7,4,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,3,7,5,4,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,3,7,5,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,3,7,6,4,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,3,7,6,5,4] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7] => ? = 3 - 1
[1,2,4,3,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 4 - 1
[1,2,4,3,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 3 - 1
[1,2,4,3,6,5,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 2 - 1
[1,2,4,3,6,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 3 - 1
[1,2,4,3,7,5,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 2 - 1
[1,2,4,3,7,6,5] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 2 - 1
[1,2,4,5,3,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,4,5,3,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,4,5,6,3,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,4,5,6,7,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 5 - 1
[1,2,4,5,7,3,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,4,5,7,6,3] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 4 - 1
[1,2,4,6,3,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,4,6,3,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,4,6,5,3,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,4,6,5,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,4,6,7,3,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,4,6,7,5,3] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 4 - 1
[1,2,4,7,3,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,4,7,3,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,4,7,5,3,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,4,7,5,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,4,7,6,3,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,4,7,6,5,3] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7] => ? = 3 - 1
[1,2,5,3,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 4 - 1
[1,2,5,3,4,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 3 - 1
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St001727
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 83%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,2] => [2] => [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 1 = 2 - 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 2 = 3 - 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1 = 2 - 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0 = 1 - 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 3 = 4 - 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2 = 3 - 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2 = 3 - 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1 = 2 - 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2 = 3 - 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1 = 2 - 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[1,2,3,4,5,6,7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => ? = 6 - 1
[1,2,3,4,5,7,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 5 - 1
[1,2,3,4,6,5,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,3,4,6,7,5] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 5 - 1
[1,2,3,4,7,5,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,3,4,7,6,5] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 4 - 1
[1,2,3,5,4,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,3,5,4,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,3,5,6,4,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,3,5,6,7,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 5 - 1
[1,2,3,5,7,4,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,3,5,7,6,4] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 4 - 1
[1,2,3,6,4,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,3,6,4,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,3,6,5,4,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,3,6,5,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,3,6,7,4,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,3,6,7,5,4] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 4 - 1
[1,2,3,7,4,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,3,7,4,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,3,7,5,4,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,3,7,5,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,3,7,6,4,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,3,7,6,5,4] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7] => ? = 3 - 1
[1,2,4,3,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 4 - 1
[1,2,4,3,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 3 - 1
[1,2,4,3,6,5,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 2 - 1
[1,2,4,3,6,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 3 - 1
[1,2,4,3,7,5,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 2 - 1
[1,2,4,3,7,6,5] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 2 - 1
[1,2,4,5,3,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,4,5,3,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,4,5,6,3,7] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,4,5,6,7,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 5 - 1
[1,2,4,5,7,3,6] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,4,5,7,6,3] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 4 - 1
[1,2,4,6,3,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,4,6,3,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,4,6,5,3,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,4,6,5,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,4,6,7,3,5] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 4 - 1
[1,2,4,6,7,5,3] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 4 - 1
[1,2,4,7,3,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 4 - 1
[1,2,4,7,3,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,4,7,5,3,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,4,7,5,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 3 - 1
[1,2,4,7,6,3,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 3 - 1
[1,2,4,7,6,5,3] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7] => ? = 3 - 1
[1,2,5,3,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,1,2,7,4,5,6] => ? = 4 - 1
[1,2,5,3,4,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 3 - 1
Description
The number of invisible inversions of a permutation.
A visible inversion of a permutation $\pi$ is a pair $i < j$ such that $\pi(j) \leq \min(i, \pi(i))$. Thus, an invisible inversion satisfies $\pi(i) > \pi(j) > i$.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000732The number of double deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St000731The number of double exceedences of a permutation. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000317The cycle descent number of a permutation. St000837The number of ascents of distance 2 of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001082The number of boxed occurrences of 123 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001651The Frankl number of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001875The number of simple modules with projective dimension at most 1.
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