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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000118
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(load all 10 compositions to match this statistic)
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> 0
[.,[.,.]]
=> [[.,.],.]
=> 0
[[.,.],.]
=> [.,[.,.]]
=> 0
[.,[.,[.,.]]]
=> [[[.,.],.],.]
=> 0
[.,[[.,.],.]]
=> [[.,[.,.]],.]
=> 0
[[.,.],[.,.]]
=> [[.,.],[.,.]]
=> 0
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> 0
[[[.,.],.],.]
=> [.,[.,[.,.]]]
=> 1
[.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> 0
[.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> 0
[.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> 0
[.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> 0
[.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> 1
[[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> 0
[[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> 0
[[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> 0
[[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> 1
[[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> 0
[[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> 0
[[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> 1
[[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> 1
[[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> 2
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [[.,[.,[[.,.],.]]],.]
=> 1
[.,[[[[.,.],.],.],.]]
=> [[.,[.,[.,[.,.]]]],.]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [[[[.,.],.],.],[.,.]]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [[[.,[.,.]],.],[.,.]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [[[.,.],[.,.]],[.,.]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 0
[[.,.],[[[.,.],.],.]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [[[.,.],.],[[.,.],.]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [[.,[.,.]],[[.,.],.]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,.]]]
=> 1
[[[.,.],.],[[.,.],.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,.],[[[.,.],.],.]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [[.,.],[[.,[.,.]],.]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [[.,.],[[.,.],[.,.]]]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [[.,.],[.,[[.,.],.]]]
=> 1
[[[[.,.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree.
[[oeis:A001006]] counts binary trees avoiding this pattern.
Matching statistic: St001066
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(load all 10 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001483
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ 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%
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
[.,.]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
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: St000931
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000931: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[[.,.],.]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 0
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
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 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 0
[[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 0
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 0
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 0
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [7,3,4,5,6,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [6,3,4,5,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [4,5,6,7,2,3,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 2
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [6,7,3,4,2,5,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [6,7,4,2,3,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [6,7,2,3,4,5,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [7,2,3,4,5,6,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 1
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 2
[.,[[[.,.],[.,[[.,.],.]]],.]]
=> [5,6,4,2,3,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [6,2,3,4,5,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => [[[[[.,[.,.]],.],.],.],[.,.]]
=> [2,1,3,4,5,7,6] => ? = 0
[[.,.],[.,[.,[[[.,.],.],.]]]]
=> [5,6,7,4,3,1,2] => [[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => ? = 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 0
[[.,.],[.,[[.,[[.,.],.]],.]]]
=> [5,6,4,7,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 0
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
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 0
[[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 0
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 0
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 0
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [7,3,4,5,6,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [6,3,4,5,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [4,5,6,7,2,3,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 2
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [6,7,3,4,2,5,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [6,7,4,2,3,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [6,7,2,3,4,5,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [7,2,3,4,5,6,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 1
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 2
[.,[[[.,.],[.,[[.,.],.]]],.]]
=> [5,6,4,2,3,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [6,2,3,4,5,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[.,[[[[[.,[.,.]],.],.],.],.]]
=> [3,2,4,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[[.,.],[.,[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,3,1,2] => [[[[[.,[.,.]],.],.],.],[.,.]]
=> [2,1,3,4,5,7,6] => ? = 0
[[.,.],[.,[.,[[[.,.],.],.]]]]
=> [5,6,7,4,3,1,2] => [[[[.,[.,[.,.]]],.],.],[.,.]]
=> [3,2,1,4,5,7,6] => ? = 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> [6,7,4,5,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 0
[[.,.],[.,[[.,[[.,.],.]],.]]]
=> [5,6,4,7,3,1,2] => [[[[.,[.,.]],[.,.]],.],[.,.]]
=> [2,1,4,3,5,7,6] => ? = 0
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: St000836
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 83%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 83%
Values
[.,.]
=> [1] => [.,.]
=> [1] => ? = 0
[.,[.,.]]
=> [2,1] => [[.,.],.]
=> [1,2] => 0
[[.,.],.]
=> [1,2] => [.,[.,.]]
=> [2,1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [[.,[.,.]],.]
=> [2,1,3] => 0
[[.,.],[.,.]]
=> [3,1,2] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> [1,3,2,4,5] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [2,1,4,3,5] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,4,3,2,5] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 2
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [6,7,5,4,3,2,1] => [[[[[[.,[.,.]],.],.],.],.],.]
=> [2,1,3,4,5,6,7] => ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [5,6,7,4,3,2,1] => [[[[[.,[.,[.,.]]],.],.],.],.]
=> [3,2,1,4,5,6,7] => ? = 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [6,7,4,5,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [5,6,4,7,3,2,1] => [[[[[.,[.,.]],[.,.]],.],.],.]
=> [2,1,4,3,5,6,7] => ? = 0
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [4,5,6,7,3,2,1] => [[[[.,[.,[.,[.,.]]]],.],.],.]
=> [4,3,2,1,5,6,7] => ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [6,7,5,3,4,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 0
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [5,6,7,3,4,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [6,7,4,3,5,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 0
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [6,7,3,4,5,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [7,3,4,5,6,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [5,6,4,3,7,2,1] => [[[[[.,[.,.]],.],[.,.]],.],.]
=> [2,1,3,5,4,6,7] => ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [4,5,6,3,7,2,1] => [[[[.,[.,[.,.]]],[.,.]],.],.]
=> [3,2,1,5,4,6,7] => ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [5,6,3,4,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [6,3,4,5,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [4,5,3,6,7,2,1] => [[[[.,[.,.]],[.,[.,.]]],.],.]
=> [2,1,5,4,3,6,7] => ? = 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [5,3,4,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [4,3,5,6,7,2,1] => [[[[.,.],[.,[.,[.,.]]]],.],.]
=> [1,5,4,3,2,6,7] => ? = 2
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [3,4,5,6,7,2,1] => [[[.,[.,[.,[.,[.,.]]]]],.],.]
=> [5,4,3,2,1,6,7] => ? = 3
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [6,7,5,4,2,3,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [5,6,7,4,2,3,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [6,7,4,5,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [5,6,4,7,2,3,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [4,5,6,7,2,3,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 2
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [6,7,5,3,2,4,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [5,6,7,3,2,4,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [6,7,5,2,3,4,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [5,6,7,2,3,4,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [6,7,4,3,2,5,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [6,7,3,4,2,5,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [6,7,4,2,3,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [6,7,3,2,4,5,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [6,7,2,3,4,5,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [7,2,3,4,5,6,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [5,6,4,3,2,7,1] => [[[[[.,[.,.]],.],.],[.,.]],.]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,[.,[[[.,.],.],.]]],.]]
=> [4,5,6,3,2,7,1] => [[[[.,[.,[.,.]]],.],[.,.]],.]
=> [3,2,1,4,6,5,7] => ? = 1
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [5,6,3,4,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,[[.,[[.,.],.]],.]],.]]
=> [4,5,3,6,2,7,1] => [[[[.,[.,.]],[.,.]],[.,.]],.]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,[[[[.,.],.],.],.]],.]]
=> [3,4,5,6,2,7,1] => [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [4,3,2,1,6,5,7] => ? = 2
[.,[[[.,.],[.,[[.,.],.]]],.]]
=> [5,6,4,2,3,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,.],[[[.,.],.],.]],.]]
=> [4,5,6,2,3,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[[.,[.,.]],[[.,.],.]],.]]
=> [5,6,3,2,4,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[[.,.],.],[[.,.],.]],.]]
=> [5,6,2,3,4,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],.],[.,.]],.]]
=> [6,2,3,4,5,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[.,[[[.,[.,[[.,.],.]]],.],.]]
=> [4,5,3,2,6,7,1] => [[[[.,[.,.]],.],[.,[.,.]]],.]
=> [2,1,3,6,5,4,7] => ? = 1
[.,[[[.,[[[.,.],.],.]],.],.]]
=> [3,4,5,2,6,7,1] => [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [3,2,1,6,5,4,7] => ? = 2
[.,[[[[.,.],[[.,.],.]],.],.]]
=> [4,5,2,3,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],.],[.,.]],.],.]]
=> [5,2,3,4,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
[.,[[[[.,[[.,.],.]],.],.],.]]
=> [3,4,2,5,6,7,1] => [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [2,1,6,5,4,3,7] => ? = 2
[.,[[[[[.,.],[.,.]],.],.],.]]
=> [4,2,3,5,6,7,1] => [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,6,5,4,3,2,7] => ? = 3
Description
The number of descents of distance 2 of a permutation.
This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St000731
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1] => 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => 0
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 0
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 0
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 0
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 0
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 2
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,4,2,5,6,7] => ? = 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,5,1,2,4,6,7] => ? = 0
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,1,2,3,6,7] => ? = 0
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,4,5,1,3,6,7] => ? = 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,3,4,6,7] => ? = 0
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [4,1,5,2,3,6,7] => ? = 0
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,1,2,6,7] => ? = 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,1,5,3,4,6,7] => ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,5,2,4,6,7] => ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,1,4,6,7] => ? = 2
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [3,1,2,5,4,6,7] => ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,3,1,5,4,6,7] => ? = 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,1,2,5,3,6,7] => ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,4,1,5,2,6,7] => ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,3,6,7] => ? = 2
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => ? = 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,4,5,2,6,7] => ? = 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5,7] => ? = 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,5,6,2,3,4,7] => ? = 0
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,5,6,1,3,4,7] => ? = 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [2,1,5,6,3,4,7] => ? = 0
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,5,6,2,4,7] => ? = 1
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,6,1,4,7] => ? = 2
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5,7] => ? = 0
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,5,2,6,3,4,7] => ? = 0
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [2,5,1,6,3,4,7] => ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5,7] => ? = 1
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,1,5,6,2,3,7] => ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,1,2,7] => ? = 2
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [3,5,1,6,2,4,7] => ? = 0
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [3,1,6,2,4,5,7] => ? = 1
[.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => ? = 2
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,4,5,6,1,3,7] => ? = 2
[.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,1,3,6,4,5,7] => ? = 0
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,1,2,6,4,5,7] => ? = 0
[.,[[.,[[[.,.],.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,3,1,6,4,5,7] => ? = 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5,7] => ? = 1
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5,7] => ? = 1
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5,7] => ? = 2
[.,[[[.,[[.,.],.]],.],[.,.]]]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,3,5,7] => ? = 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5,7] => ? = 2
[.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5,7] => ? = 3
[.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,[.,[[.,.],[.,.]]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [3,1,2,4,6,5,7] => ? = 0
[.,[[.,[.,[[[.,.],.],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,3,1,4,6,5,7] => ? = 1
[.,[[.,[[.,.],[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [4,1,2,3,6,5,7] => ? = 0
[.,[[.,[[.,.],[[.,.],.]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [3,4,1,2,6,5,7] => ? = 0
[.,[[.,[[[.,.],.],[.,.]]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => ? = 1
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St001744
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 83%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 83%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1] => 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> [2,1] => 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 0
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 2
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => ? = 0
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7] => ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7] => ? = 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 0
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,1,5,6,7] => ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5,6,7] => ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,4,1,3,5,6,7] => ? = 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [3,4,1,2,5,6,7] => ? = 1
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7] => ? = 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 0
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,6,7] => ? = 0
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => ? = 0
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,3,4,6,7] => ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,1,6,7] => ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2,6,7] => ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,1,5,3,6,7] => ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [3,4,1,5,2,6,7] => ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,1,2,5,3,6,7] => ? = 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,5,1,4,6,7] => ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [3,1,5,2,4,6,7] => ? = 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [2,4,5,1,3,6,7] => ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,5,1,3,4,6,7] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [3,4,5,1,2,6,7] => ? = 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3,6,7] => ? = 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [3,5,1,2,4,6,7] => ? = 2
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [4,5,1,2,3,6,7] => ? = 2
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 3
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 0
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [2,3,1,4,6,5,7] => ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => ? = 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5,7] => ? = 0
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [3,1,4,2,6,5,7] => ? = 0
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [2,4,1,3,6,5,7] => ? = 1
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [3,4,1,2,6,5,7] => ? = 1
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 2
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,6,4,7] => ? = 0
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1,5,6,4,7] => ? = 0
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [3,1,2,5,6,4,7] => ? = 1
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 1
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1,6,4,5,7] => ? = 1
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 2
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => ? = 0
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,6,4,7] => ? = 0
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,3,5,7] => ? = 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [2,1,5,6,3,4,7] => ? = 1
[.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,3,4,5,7] => ? = 2
[.,[[.,[[[.,.],.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,2,3,6,4,7] => ? = 1
Description
The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Matching statistic: St000365
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1] => 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 0
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 0
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 0
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 0
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 0
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => 0
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => ? = 0
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? = 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => ? = 0
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 0
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => ? = 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5] => ? = 0
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => ? = 0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,4,3,2,7,6,1] => ? = 0
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,3,2,1,5,7,6] => ? = 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,5,4,3,7,2,1] => ? = 0
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [4,3,2,1,6,7,5] => ? = 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ? = 1
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => ? = 2
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,2,1,7,6,5,4] => ? = 0
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,2,1,6,5,4,7] => ? = 0
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,5,4,7,6] => ? = 0
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,2,1,6,5,7,4] => ? = 0
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,1] => ? = 0
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [4,3,2,6,5,1,7] => ? = 0
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => ? = 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,2,1,4,6,5,7] => ? = 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [5,4,3,7,6,2,1] => ? = 0
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [4,3,2,5,1,7,6] => ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,2,1,5,7,6,4] => ? = 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [4,3,2,5,7,6,1] => ? = 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,2,1,4,5,7,6] => ? = 2
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [6,5,4,7,3,2,1] => ? = 0
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,4,3,6,2,1,7] => ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [4,3,2,6,5,7,1] => ? = 0
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,4,3,6,2,7,1] => ? = 0
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,3,2,5,1,6,7] => ? = 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,2,1,6,7,5,4] => ? = 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [3,2,1,5,6,4,7] => ? = 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [4,3,2,6,7,5,1] => ? = 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [3,2,1,4,6,7,5] => ? = 2
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => ? = 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [4,3,2,5,6,1,7] => ? = 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [3,2,1,5,6,7,4] => ? = 2
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ? = 2
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7] => ? = 3
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,6,5,4,3] => ? = 0
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ? = 0
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,5,4,3,7,6] => ? = 0
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,6,5,4,7,3] => ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => ? = 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => ? = 0
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 0
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
=> [2,1,5,4,7,6,3] => ? = 0
Description
The number of double ascents of a permutation.
A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000732The number of double deficiencies of a permutation. St000039The number of crossings of a permutation. St000317The cycle descent number of 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.
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