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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St000365
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
St000365: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 1
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 2
[1,2,4,3] => 1
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 1
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 1
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 3
[1,2,3,5,4] => 2
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 2
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 1
Description
The number of double ascents of a permutation.
A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
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
[1,2] => [.,[.,.]]
=> [.,[.,.]]
=> 0
[2,1] => [[.,.],.]
=> [[.,.],.]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> 0
[3,1,2] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [[[.,.],.],.]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> 0
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> 3
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> 0
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> 0
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> 1
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree.
[[oeis:A001006]] counts binary trees avoiding this pattern.
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 = 0 + 1
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
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
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[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
[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
[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
[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
[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
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 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
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 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
[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
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 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
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 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
[1,2] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[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
[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
[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
[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
[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
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 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
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 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
[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
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 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
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 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: 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 = 0 + 1
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 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 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 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 = 3 + 1
[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 = 2 + 1
[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 + 1
[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 = 2 + 1
[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 + 1
[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 + 1
[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 + 1
[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 = 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 + 1
[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 = 2 + 1
[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 + 1
[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 + 1
[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 + 1
[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 = 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 = 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 = 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 = 1 + 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)
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]
=> ? = 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 0
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 2
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 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)
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => 1
[1,3,2] => [2,3,1] => [1,3,2] => 0
[2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 2
[1,2,4,3] => [3,4,2,1] => [1,4,3,2] => 1
[1,3,2,4] => [4,2,3,1] => [4,1,3,2] => 0
[1,3,4,2] => [2,4,3,1] => [1,4,3,2] => 1
[1,4,2,3] => [3,2,4,1] => [2,1,4,3] => 0
[1,4,3,2] => [2,3,4,1] => [1,2,4,3] => 0
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 1
[2,1,4,3] => [3,4,1,2] => [2,4,1,3] => 0
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 0
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 1
[2,4,1,3] => [3,1,4,2] => [2,1,4,3] => 0
[2,4,3,1] => [1,3,4,2] => [1,2,4,3] => 0
[3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 0
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 1
[4,1,3,2] => [2,3,1,4] => [1,3,2,4] => 0
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 3
[1,2,3,5,4] => [4,5,3,2,1] => [1,5,4,3,2] => 2
[1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,3,2] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,5,4,3,2] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [2,1,5,4,3] => 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,5,4,3] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,1,3,2] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,5,1,4,3] => 0
[1,3,4,2,5] => [5,2,4,3,1] => [5,1,4,3,2] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,5,4,3,2] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [2,1,5,4,3] => 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,5,4,3] => 1
[1,4,2,3,5] => [5,3,2,4,1] => [5,2,1,4,3] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,5,1,4,3] => 0
[1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [1,5,2,4,3] => 0
[1,4,5,2,3] => [3,2,5,4,1] => [2,1,5,4,3] => 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 3
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 3
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,7,1,6,5,4,3] => ? = 2
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,1,6,5,4,3,2] => ? = 3
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,2,1,6,5,4,3] => ? = 3
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [2,7,1,6,5,4,3] => ? = 2
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,1,2,6,5,4,3] => ? = 2
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,7,2,6,5,4,3] => ? = 2
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [3,2,1,7,6,5,4] => ? = 3
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [3,1,2,7,6,5,4] => ? = 2
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [2,1,3,7,6,5,4] => ? = 2
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => ? = 3
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,2,6,1,5,4,3] => ? = 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 2
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [3,2,7,1,6,5,4] => ? = 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 3
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,7,1,6,5,4,3] => ? = 2
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [7,1,6,5,4,3,2] => ? = 3
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [7,2,1,6,5,4,3] => ? = 3
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [2,7,1,6,5,4,3] => ? = 2
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [7,1,2,6,5,4,3] => ? = 2
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,7,2,6,5,4,3] => ? = 2
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [3,2,1,7,6,5,4] => ? = 3
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [3,1,2,7,6,5,4] => ? = 2
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [2,1,3,7,6,5,4] => ? = 2
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,6,2,1,5,4,3] => ? = 3
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [3,7,2,1,6,5,4] => ? = 2
[1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [7,2,6,1,5,4,3] => ? = 1
[1,2,5,3,6,7,4] => [4,7,6,3,5,2,1] => [2,7,6,1,5,4,3] => ? = 2
[1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [3,2,7,1,6,5,4] => ? = 1
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,1,2,5,4,3] => ? = 2
[1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [3,7,1,2,6,5,4] => ? = 1
[1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [7,1,6,2,5,4,3] => ? = 1
[1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [1,7,6,2,5,4,3] => ? = 2
[1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => [3,1,7,2,6,5,4] => ? = 1
[1,2,5,6,3,4,7] => [7,4,3,6,5,2,1] => [7,2,1,6,5,4,3] => ? = 3
[1,2,5,6,3,7,4] => [4,7,3,6,5,2,1] => [2,7,1,6,5,4,3] => ? = 2
[1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => [7,1,2,6,5,4,3] => ? = 2
[1,2,5,6,4,7,3] => [3,7,4,6,5,2,1] => [1,7,2,6,5,4,3] => ? = 2
[1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => [3,2,1,7,6,5,4] => ? = 3
[1,2,5,7,4,3,6] => [6,3,4,7,5,2,1] => [3,1,2,7,6,5,4] => ? = 2
[1,2,5,7,6,3,4] => [4,3,6,7,5,2,1] => [2,1,3,7,6,5,4] => ? = 2
[1,2,6,3,4,5,7] => [7,5,4,3,6,2,1] => [7,3,2,1,6,5,4] => ? = 3
[1,2,6,3,4,7,5] => [5,7,4,3,6,2,1] => [3,7,2,1,6,5,4] => ? = 2
[1,2,6,3,5,4,7] => [7,4,5,3,6,2,1] => [7,1,3,2,6,5,4] => ? = 1
[1,2,6,3,5,7,4] => [4,7,5,3,6,2,1] => [1,7,3,2,6,5,4] => ? = 2
[1,2,6,3,7,4,5] => [5,4,7,3,6,2,1] => [3,2,7,1,6,5,4] => ? = 1
[1,2,6,4,3,5,7] => [7,5,3,4,6,2,1] => [7,3,1,2,6,5,4] => ? = 2
[1,2,6,4,3,7,5] => [5,7,3,4,6,2,1] => [3,7,1,2,6,5,4] => ? = 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: St000731
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [2,3,1] => 1
[1,3,2] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [2,3,4,1] => 2
[1,2,4,3] => [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 1
[1,3,2,4] => [4,2,3,1] => [4,1,3,2] => [3,4,1,2] => 0
[1,3,4,2] => [2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 1
[1,4,2,3] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[1,4,3,2] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => [3,1,4,2] => 1
[2,1,4,3] => [3,4,1,2] => [2,4,1,3] => [4,2,1,3] => 0
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => [3,4,1,2] => 0
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 1
[2,4,1,3] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[2,4,3,1] => [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [4,2,1,3] => [4,2,1,3] => [2,4,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [4,2,1,3] => 0
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 1
[4,1,3,2] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => 3
[1,2,3,5,4] => [4,5,3,2,1] => [1,5,4,3,2] => [1,3,4,5,2] => 2
[1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,3,2] => [3,4,5,1,2] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,5,4,3,2] => [1,3,4,5,2] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [2,1,5,4,3] => [2,1,4,5,3] => 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,1,3,2] => [3,4,1,5,2] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,5,1,4,3] => [4,2,5,1,3] => 0
[1,3,4,2,5] => [5,2,4,3,1] => [5,1,4,3,2] => [3,4,5,1,2] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,5,4,3,2] => [1,3,4,5,2] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [2,1,5,4,3] => [2,1,4,5,3] => 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[1,4,2,3,5] => [5,3,2,4,1] => [5,2,1,4,3] => [2,4,5,1,3] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,5,1,4,3] => [4,2,5,1,3] => 0
[1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => [4,1,5,2,3] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [1,5,2,4,3] => [1,4,5,2,3] => 0
[1,4,5,2,3] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,4,5,3] => 1
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => ? = 3
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => [3,4,5,6,1,7,2] => ? = 3
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,7,1,6,5,4,3] => [4,2,5,6,7,1,3] => ? = 2
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => ? = 3
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,2,1,6,5,4,3] => [2,4,5,6,7,1,3] => ? = 3
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [2,7,1,6,5,4,3] => [4,2,5,6,7,1,3] => ? = 2
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,1,2,6,5,4,3] => [4,1,5,6,7,2,3] => ? = 2
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => ? = 3
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [3,2,1,7,6,5,4] => [2,3,1,5,6,7,4] => ? = 3
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [3,1,2,7,6,5,4] => [3,1,2,5,6,7,4] => ? = 2
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [2,1,3,7,6,5,4] => [2,1,3,5,6,7,4] => ? = 2
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => [3,4,5,1,6,7,2] => ? = 3
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,7,6,1,5,4,3] => [4,2,5,6,1,7,3] => ? = 2
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,2,6,1,5,4,3] => [4,5,6,7,2,1,3] => ? = 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [2,7,6,1,5,4,3] => [4,2,5,6,1,7,3] => ? = 2
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [3,2,7,1,6,5,4] => [5,3,2,6,7,1,4] => ? = 1
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,3,7,2,6,5,4] => [1,5,3,6,7,2,4] => ? = 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,1,5,4,3,2] => [3,4,5,6,1,7,2] => ? = 3
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,7,1,6,5,4,3] => [4,2,5,6,7,1,3] => ? = 2
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => ? = 3
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [7,2,1,6,5,4,3] => [2,4,5,6,7,1,3] => ? = 3
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [2,7,1,6,5,4,3] => [4,2,5,6,7,1,3] => ? = 2
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [7,1,2,6,5,4,3] => [4,1,5,6,7,2,3] => ? = 2
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => ? = 3
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [3,2,1,7,6,5,4] => [2,3,1,5,6,7,4] => ? = 3
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [3,1,2,7,6,5,4] => [3,1,2,5,6,7,4] => ? = 2
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [2,1,3,7,6,5,4] => [2,1,3,5,6,7,4] => ? = 2
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,6,2,1,5,4,3] => [2,4,5,6,1,7,3] => ? = 3
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [3,7,2,1,6,5,4] => [2,5,3,6,7,1,4] => ? = 2
[1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [7,2,6,1,5,4,3] => [4,5,6,7,2,1,3] => ? = 1
[1,2,5,3,6,7,4] => [4,7,6,3,5,2,1] => [2,7,6,1,5,4,3] => [4,2,5,6,1,7,3] => ? = 2
[1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [3,2,7,1,6,5,4] => [5,3,2,6,7,1,4] => ? = 1
[1,2,5,3,7,6,4] => [4,6,7,3,5,2,1] => [1,3,7,2,6,5,4] => [1,5,3,6,7,2,4] => ? = 1
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,1,2,5,4,3] => [4,1,5,6,2,7,3] => ? = 2
[1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [3,7,1,2,6,5,4] => [5,1,3,6,7,2,4] => ? = 1
[1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [7,1,6,2,5,4,3] => [4,5,6,7,1,2,3] => ? = 1
[1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => [3,1,7,2,6,5,4] => [5,3,1,6,7,2,4] => ? = 1
[1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => [1,3,7,2,6,5,4] => [1,5,3,6,7,2,4] => ? = 1
[1,2,5,6,3,4,7] => [7,4,3,6,5,2,1] => [7,2,1,6,5,4,3] => [2,4,5,6,7,1,3] => ? = 3
[1,2,5,6,3,7,4] => [4,7,3,6,5,2,1] => [2,7,1,6,5,4,3] => [4,2,5,6,7,1,3] => ? = 2
[1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => [7,1,2,6,5,4,3] => [4,1,5,6,7,2,3] => ? = 2
[1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => ? = 3
[1,2,5,7,3,4,6] => [6,4,3,7,5,2,1] => [3,2,1,7,6,5,4] => [2,3,1,5,6,7,4] => ? = 3
[1,2,5,7,4,3,6] => [6,3,4,7,5,2,1] => [3,1,2,7,6,5,4] => [3,1,2,5,6,7,4] => ? = 2
[1,2,5,7,6,3,4] => [4,3,6,7,5,2,1] => [2,1,3,7,6,5,4] => [2,1,3,5,6,7,4] => ? = 2
[1,2,6,3,4,7,5] => [5,7,4,3,6,2,1] => [3,7,2,1,6,5,4] => [2,5,3,6,7,1,4] => ? = 2
[1,2,6,3,5,4,7] => [7,4,5,3,6,2,1] => [7,1,3,2,6,5,4] => [3,5,1,6,7,2,4] => ? = 1
[1,2,6,3,7,4,5] => [5,4,7,3,6,2,1] => [3,2,7,1,6,5,4] => [5,3,2,6,7,1,4] => ? = 1
[1,2,6,3,7,5,4] => [4,5,7,3,6,2,1] => [1,3,7,2,6,5,4] => [1,5,3,6,7,2,4] => ? = 1
[1,2,6,4,3,5,7] => [7,5,3,4,6,2,1] => [7,3,1,2,6,5,4] => [3,1,5,6,7,2,4] => ? = 2
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: St000732
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [2,1] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [3,1,2] => 1
[1,3,2] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [3,1,2] => [3,1,2] => [2,3,1] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [4,1,2,3] => 2
[1,2,4,3] => [3,4,2,1] => [1,4,3,2] => [1,4,2,3] => 1
[1,3,2,4] => [4,2,3,1] => [4,1,3,2] => [3,4,2,1] => 0
[1,3,4,2] => [2,4,3,1] => [1,4,3,2] => [1,4,2,3] => 1
[1,4,2,3] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[1,4,3,2] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => [2,4,1,3] => 1
[2,1,4,3] => [3,4,1,2] => [2,4,1,3] => [3,2,4,1] => 0
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => [3,4,2,1] => 0
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 1
[2,4,1,3] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[2,4,3,1] => [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [4,2,1,3] => [4,2,1,3] => [3,1,4,2] => 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [3,2,4,1] => 0
[3,2,1,4] => [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 0
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [3,1,2,4] => 1
[4,1,3,2] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,1,3] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [5,1,2,3,4] => 3
[1,2,3,5,4] => [4,5,3,2,1] => [1,5,4,3,2] => [1,5,2,3,4] => 2
[1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,3,2] => [4,5,2,3,1] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,5,4,3,2] => [1,5,2,3,4] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [2,1,5,4,3] => [2,1,5,3,4] => 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,1,3,2] => [3,5,2,1,4] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,5,1,4,3] => [4,2,5,3,1] => 0
[1,3,4,2,5] => [5,2,4,3,1] => [5,1,4,3,2] => [4,5,2,3,1] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,5,4,3,2] => [1,5,2,3,4] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [2,1,5,4,3] => [2,1,5,3,4] => 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,5,4,3] => [1,2,5,3,4] => 1
[1,4,2,3,5] => [5,3,2,4,1] => [5,2,1,4,3] => [4,1,5,3,2] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,5,1,4,3] => [4,2,5,3,1] => 0
[1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => [2,4,5,3,1] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [1,5,2,4,3] => [1,4,5,3,2] => 0
[1,4,5,2,3] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,3,4] => 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,2,5,4,3] => [1,2,5,3,4] => 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,1,2,3,4,5,6] => ? = 5
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 4
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,1,6,5,4,3,2] => [6,7,2,3,4,5,1] => ? = 3
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 4
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => [2,1,7,3,4,5,6] => ? = 3
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => [5,7,2,3,4,1,6] => ? = 3
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,7,1,6,5,4,3] => [6,2,7,3,4,5,1] => ? = 2
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,1,6,5,4,3,2] => [6,7,2,3,4,5,1] => ? = 3
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 4
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,1,7,6,5,4,3] => [2,1,7,3,4,5,6] => ? = 3
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,2,1,6,5,4,3] => [6,1,7,3,4,5,2] => ? = 3
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [2,7,1,6,5,4,3] => [6,2,7,3,4,5,1] => ? = 2
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,1,2,6,5,4,3] => [2,6,7,3,4,5,1] => ? = 2
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,7,2,6,5,4,3] => [1,6,7,3,4,5,2] => ? = 2
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [2,1,7,6,5,4,3] => [2,1,7,3,4,5,6] => ? = 3
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [3,2,1,7,6,5,4] => [3,1,2,7,4,5,6] => ? = 3
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [3,1,2,7,6,5,4] => [2,3,1,7,4,5,6] => ? = 2
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [2,1,3,7,6,5,4] => [2,1,3,7,4,5,6] => ? = 2
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => [4,7,2,3,1,5,6] => ? = 3
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,7,6,1,5,4,3] => [5,2,7,3,4,1,6] => ? = 2
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,2,6,1,5,4,3] => [5,6,7,3,4,1,2] => ? = 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [2,7,6,1,5,4,3] => [5,2,7,3,4,1,6] => ? = 2
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [3,2,7,1,6,5,4] => [6,3,2,7,4,5,1] => ? = 1
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,3,7,2,6,5,4] => [1,6,3,7,4,5,2] => ? = 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,1,5,4,3,2] => [5,7,2,3,4,1,6] => ? = 3
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,7,1,6,5,4,3] => [6,2,7,3,4,5,1] => ? = 2
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [7,1,6,5,4,3,2] => [6,7,2,3,4,5,1] => ? = 3
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 4
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [2,1,7,6,5,4,3] => [2,1,7,3,4,5,6] => ? = 3
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [7,2,1,6,5,4,3] => [6,1,7,3,4,5,2] => ? = 3
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [2,7,1,6,5,4,3] => [6,2,7,3,4,5,1] => ? = 2
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [7,1,2,6,5,4,3] => [2,6,7,3,4,5,1] => ? = 2
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,7,2,6,5,4,3] => [1,6,7,3,4,5,2] => ? = 2
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [2,1,7,6,5,4,3] => [2,1,7,3,4,5,6] => ? = 3
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [3,2,1,7,6,5,4] => [3,1,2,7,4,5,6] => ? = 3
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [3,1,2,7,6,5,4] => [2,3,1,7,4,5,6] => ? = 2
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [2,1,3,7,6,5,4] => [2,1,3,7,4,5,6] => ? = 2
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,6,2,1,5,4,3] => [5,1,7,3,4,2,6] => ? = 3
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [3,7,2,1,6,5,4] => [6,1,3,7,4,5,2] => ? = 2
[1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [7,2,6,1,5,4,3] => [5,6,7,3,4,1,2] => ? = 1
[1,2,5,3,6,7,4] => [4,7,6,3,5,2,1] => [2,7,6,1,5,4,3] => [5,2,7,3,4,1,6] => ? = 2
[1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [3,2,7,1,6,5,4] => [6,3,2,7,4,5,1] => ? = 1
[1,2,5,3,7,6,4] => [4,6,7,3,5,2,1] => [1,3,7,2,6,5,4] => [1,6,3,7,4,5,2] => ? = 1
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,1,2,5,4,3] => [2,5,7,3,4,1,6] => ? = 2
[1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [3,7,1,2,6,5,4] => [2,6,3,7,4,5,1] => ? = 1
[1,2,5,4,6,3,7] => [7,3,6,4,5,2,1] => [7,1,6,2,5,4,3] => [6,5,7,3,4,2,1] => ? = 1
[1,2,5,4,6,7,3] => [3,7,6,4,5,2,1] => [1,7,6,2,5,4,3] => [1,5,7,3,4,2,6] => ? = 2
[1,2,5,4,7,3,6] => [6,3,7,4,5,2,1] => [3,1,7,2,6,5,4] => [3,6,1,7,4,5,2] => ? = 1
[1,2,5,4,7,6,3] => [3,6,7,4,5,2,1] => [1,3,7,2,6,5,4] => [1,6,3,7,4,5,2] => ? = 1
Description
The number of double deficiencies of a permutation.
A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.
The following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000371The number of mid points of decreasing subsequences of length 3 in 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. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. 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. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 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. St000836The number of descents of distance 2 of a permutation. 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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