searching the database
Your data matches 13 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000441
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00018: Binary trees —left border symmetry⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [.,.]
=> [1] => 0
[1,2] => [.,[.,.]]
=> [.,[.,.]]
=> [2,1] => 0
[2,1] => [[.,.],.]
=> [[.,.],.]
=> [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 0
[1,3,2] => [.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [2,3,1] => 1
[2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[2,3,1] => [[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [3,1,2] => 1
[3,1,2] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [2,1,3] => 0
[3,2,1] => [[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,2,3] => 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 0
Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000932
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 80%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [1] => [1,0]
=> ? = 0
[1,2] => [2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[1,2,3] => [3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[2,3,1] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[3,2,1] => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,4,3,2] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[2,1,3,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,1,4,3] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,3,1,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[2,3,4,1] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[2,4,3,1] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[3,1,2,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[3,1,4,2] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,2,1,4] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[4,1,2,3] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4,1,3,2] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,2,1,3] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,3,1,2] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,2,4,5,3] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,2,5,4,3] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,3,2,5,4] => [2,2,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,4,5,2] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,5,4,2] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,4,2,5,3] => [2,2,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [2,2,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,4,5,3,2] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[7,8,2,3,4,5,6,1] => [2,5,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[6,7,2,3,4,5,8,1] => [2,5,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[4,5,2,3,6,7,8,1] => [2,5,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[3,4,2,5,6,7,8,1] => [2,5,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[7,8,3,4,5,6,1,2] => [2,4,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[5,6,3,4,7,8,1,2] => [2,4,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[4,5,3,6,7,8,1,2] => [2,4,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[5,6,7,8,1,2,3,4] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[7,8,2,3,4,5,1,6] => [2,4,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[7,8,1,2,3,4,5,6] => [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[8,1,2,3,4,5,6,7] => [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[6,7,2,3,4,5,1,8] => [2,4,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[4,5,2,3,6,7,1,8] => [2,4,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[3,4,2,5,6,7,1,8] => [2,4,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[4,5,6,7,1,2,3,8] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[6,7,1,2,3,4,5,8] => [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[7,5,4,3,1,2,6,8] => ? => ? => ?
=> ? = 3
[7,1,2,3,4,5,6,8] => [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[3,4,5,6,1,2,7,8] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[6,1,2,3,4,5,7,8] => [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[2,3,4,5,1,6,7,8] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[4,5,1,2,3,6,7,8] => [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[5,1,2,3,4,6,7,8] => [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[3,4,1,2,5,6,7,8] => [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[4,1,2,3,5,6,7,8] => [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[2,3,1,4,5,6,7,8] => [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[3,1,2,4,5,6,7,8] => [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[2,1,3,4,5,6,7,8] => [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,2,3,4,5,6,7,8] => [8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[2,1,5,8,7,6,4,3] => [1,3,1,1,1,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[2,1,4,5,6,8,7,3] => [1,5,1,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2
[2,1,3,4,6,7,8,5] => [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[2,1,3,4,5,8,7,6] => [1,5,1,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2
[2,1,3,4,5,6,8,7] => [1,6,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,3,2,4,5,7,6,8] => [2,4,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[2,1,3,4,5,7,6,8] => [1,5,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[2,1,3,4,6,5,7,8] => [1,4,3] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
[1,2,3,5,4,6,7,8] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,3,2,4,5,6,7,8] => [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[2,6,3,4,5,1,7,8] => [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 0
[2,4,6,8,1,3,5,7] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[2,4,6,7,1,3,5,8] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[2,4,7,8,1,3,5,6] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[2,4,1,3,5,6,7,8] => [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[2,4,5,6,1,3,7,8] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[2,5,6,8,1,3,4,7] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[2,5,7,8,1,3,4,6] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[2,3,5,7,1,4,6,8] => [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[2,1,3,5,8,4,6,7] => [1,4,3] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 0
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001484
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●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]
=> [1]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0
[4,3,7,6,5,2,1] => [[[[.,.],[[[.,.],.],.]],.],.]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 4
[4,5,3,2,1,7,6] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[4,5,3,2,7,6,1] => [[[[.,[.,.]],.],[[.,.],.]],.]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 3
[4,6,3,2,1,7,5] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[4,6,3,2,7,5,1] => [[[[.,[.,.]],.],[[.,.],.]],.]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 3
[4,7,3,2,1,6,5] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[4,7,3,2,6,5,1] => [[[[.,[.,.]],.],[[.,.],.]],.]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 3
[4,7,6,5,3,2,1] => [[[[.,[[[.,.],.],.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 5
[5,3,7,6,4,2,1] => [[[[.,.],[[[.,.],.],.]],.],.]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 4
[5,4,3,2,7,6,1] => [[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 4
[5,4,7,6,3,2,1] => [[[[[.,.],[[.,.],.]],.],.],.]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 4
[5,6,3,2,1,7,4] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[5,6,3,2,7,4,1] => [[[[.,[.,.]],.],[[.,.],.]],.]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 3
[5,6,4,2,1,7,3] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[5,6,4,2,7,3,1] => [[[[.,[.,.]],.],[[.,.],.]],.]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 3
[5,6,4,3,1,7,2] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[5,7,3,2,1,6,4] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[5,7,3,2,6,4,1] => [[[[.,[.,.]],.],[[.,.],.]],.]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 3
[5,7,4,2,1,6,3] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[5,7,4,2,6,3,1] => [[[[.,[.,.]],.],[[.,.],.]],.]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 3
[5,7,4,3,1,6,2] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[6,3,7,5,4,2,1] => [[[[.,.],[[[.,.],.],.]],.],.]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 4
[6,4,3,2,7,5,1] => [[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 4
[6,4,7,5,3,2,1] => [[[[[.,.],[[.,.],.]],.],.],.]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 4
[6,5,3,2,7,4,1] => [[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 4
[6,5,4,2,7,3,1] => [[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 4
[6,7,3,2,1,5,4] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[6,7,3,2,5,4,1] => [[[[.,[.,.]],.],[[.,.],.]],.]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 3
[6,7,4,2,1,5,3] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[6,7,4,2,5,3,1] => [[[[.,[.,.]],.],[[.,.],.]],.]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 3
[6,7,4,3,1,5,2] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[6,7,5,2,1,4,3] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[6,7,5,2,4,3,1] => [[[[.,[.,.]],.],[[.,.],.]],.]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2]
=> ? = 3
[6,7,5,3,1,4,2] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[6,7,5,4,1,3,2] => [[[[.,[.,.]],.],.],[[.,.],.]]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2]
=> ? = 3
[7,3,6,5,4,2,1] => [[[[.,.],[[[.,.],.],.]],.],.]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 4
[7,4,3,2,6,5,1] => [[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 4
[7,4,6,5,3,2,1] => [[[[[.,.],[[.,.],.]],.],.],.]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 4
[7,5,3,2,6,4,1] => [[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 4
[7,5,4,2,6,3,1] => [[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 4
[7,6,3,2,5,4,1] => [[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 4
[7,6,4,2,5,3,1] => [[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 4
[7,6,5,2,4,3,1] => [[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 4
[8,6,7,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,1]
=> ? = 5
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,1]
=> ? = 5
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? = 5
[8,7,5,6,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2,1]
=> ? = 5
[8,6,5,7,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2,1]
=> ? = 5
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2,1]
=> ? = 5
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 4
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St001091
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1],[]]
=> [1]
=> 0
[1,2] => [2] => [[2],[]]
=> [2]
=> 0
[2,1] => [1,1] => [[1,1],[]]
=> [1,1]
=> 1
[1,2,3] => [3] => [[3],[]]
=> [3]
=> 0
[1,3,2] => [2,1] => [[2,2],[1]]
=> [2,2]
=> 1
[2,1,3] => [1,2] => [[2,1],[]]
=> [2,1]
=> 0
[2,3,1] => [2,1] => [[2,2],[1]]
=> [2,2]
=> 1
[3,1,2] => [1,2] => [[2,1],[]]
=> [2,1]
=> 0
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> 2
[1,2,3,4] => [4] => [[4],[]]
=> [4]
=> 0
[1,2,4,3] => [3,1] => [[3,3],[2]]
=> [3,3]
=> 1
[1,3,2,4] => [2,2] => [[3,2],[1]]
=> [3,2]
=> 0
[1,3,4,2] => [3,1] => [[3,3],[2]]
=> [3,3]
=> 1
[1,4,2,3] => [2,2] => [[3,2],[1]]
=> [3,2]
=> 0
[1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
[2,1,3,4] => [1,3] => [[3,1],[]]
=> [3,1]
=> 0
[2,1,4,3] => [1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> 1
[2,3,1,4] => [2,2] => [[3,2],[1]]
=> [3,2]
=> 0
[2,3,4,1] => [3,1] => [[3,3],[2]]
=> [3,3]
=> 1
[2,4,1,3] => [2,2] => [[3,2],[1]]
=> [3,2]
=> 0
[2,4,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
[3,1,2,4] => [1,3] => [[3,1],[]]
=> [3,1]
=> 0
[3,1,4,2] => [1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> 1
[3,2,1,4] => [1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> 1
[3,2,4,1] => [1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> 1
[3,4,1,2] => [2,2] => [[3,2],[1]]
=> [3,2]
=> 0
[3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> 2
[4,1,2,3] => [1,3] => [[3,1],[]]
=> [3,1]
=> 0
[4,1,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> 1
[4,2,1,3] => [1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> 1
[4,2,3,1] => [1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> 1
[4,3,1,2] => [1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> 1
[4,3,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> 3
[1,2,3,4,5] => [5] => [[5],[]]
=> [5]
=> 0
[1,2,3,5,4] => [4,1] => [[4,4],[3]]
=> [4,4]
=> 1
[1,2,4,3,5] => [3,2] => [[4,3],[2]]
=> [4,3]
=> 0
[1,2,4,5,3] => [4,1] => [[4,4],[3]]
=> [4,4]
=> 1
[1,2,5,3,4] => [3,2] => [[4,3],[2]]
=> [4,3]
=> 0
[1,2,5,4,3] => [3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> 2
[1,3,2,4,5] => [2,3] => [[4,2],[1]]
=> [4,2]
=> 0
[1,3,2,5,4] => [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
[1,3,4,2,5] => [3,2] => [[4,3],[2]]
=> [4,3]
=> 0
[1,3,4,5,2] => [4,1] => [[4,4],[3]]
=> [4,4]
=> 1
[1,3,5,2,4] => [3,2] => [[4,3],[2]]
=> [4,3]
=> 0
[1,3,5,4,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> 2
[1,4,2,3,5] => [2,3] => [[4,2],[1]]
=> [4,2]
=> 0
[1,4,2,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
[1,4,3,2,5] => [2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> 1
[1,4,3,5,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> 1
[1,4,5,2,3] => [3,2] => [[4,3],[2]]
=> [4,3]
=> 0
[1,3,2,7,6,5,4] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[1,4,2,7,6,5,3] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[1,4,3,2,7,6,5] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,5,2,7,6,4,3] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[1,5,3,2,7,6,4] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,5,4,2,7,6,3] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,5,4,3,7,6,2] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,6,2,7,5,4,3] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[1,6,3,2,7,5,4] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,6,4,2,7,5,3] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,6,4,3,7,5,2] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,6,5,2,7,4,3] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,6,5,3,7,4,2] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,6,5,4,7,3,2] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,6,5,7,4,3,2] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[1,7,2,6,5,4,3] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[1,7,3,2,6,5,4] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,7,4,2,6,5,3] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,7,4,3,6,5,2] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,7,5,2,6,4,3] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,7,5,3,6,4,2] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,7,5,4,6,3,2] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,7,5,6,4,3,2] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[1,7,6,2,5,4,3] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,7,6,3,5,4,2] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[1,7,6,4,5,3,2] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[2,1,3,4,7,6,5] => [1,4,1,1] => [[4,4,4,1],[3,3]]
=> [4,4,4,1]
=> ? = 2
[2,1,3,6,7,5,4] => [1,4,1,1] => [[4,4,4,1],[3,3]]
=> [4,4,4,1]
=> ? = 2
[2,3,1,7,6,5,4] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[2,4,1,7,6,5,3] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[2,4,5,6,7,3,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 2
[2,4,6,5,7,3,1] => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [4,4,4,3]
=> ? = 2
[2,4,6,7,5,3,1] => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> ? = 3
[2,4,7,5,6,3,1] => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> [4,4,4,3]
=> ? = 2
[2,5,1,7,6,4,3] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[2,6,1,7,5,4,3] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[2,7,1,6,5,4,3] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[3,1,2,4,7,6,5] => [1,4,1,1] => [[4,4,4,1],[3,3]]
=> [4,4,4,1]
=> ? = 2
[3,1,2,6,7,5,4] => [1,4,1,1] => [[4,4,4,1],[3,3]]
=> [4,4,4,1]
=> ? = 2
[3,2,5,7,6,4,1] => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [3,3,3,3,1]
=> ? = 3
[3,4,1,7,6,5,2] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[3,4,2,1,7,6,5] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[3,4,5,6,7,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [5,5,5]
=> ? = 2
[3,5,1,7,6,4,2] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[3,5,2,1,7,6,4] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[3,5,4,2,7,6,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
[3,5,6,7,4,2,1] => [4,1,1,1] => [[4,4,4,4],[3,3,3]]
=> [4,4,4,4]
=> ? = 3
[3,5,7,6,4,2,1] => [3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> [3,3,3,3,3]
=> ? = 4
[3,6,1,7,5,4,2] => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [3,3,3,3,2]
=> ? = 3
[3,6,2,1,7,5,4] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [3,3,3,2,2]
=> ? = 3
Description
The number of parts in an integer partition whose next smaller part has the same size.
In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St000445
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,1,0,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[6,7,8,5,4,3,2,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[8,7,5,6,4,3,2,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[8,6,5,7,4,3,2,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[7,6,5,8,4,3,2,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[5,6,7,8,4,3,2,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[4,5,6,7,8,3,2,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 3
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 4
[7,8,6,4,3,5,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 4
[7,8,5,4,3,6,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 4
[6,7,5,4,3,8,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 4
[7,8,6,5,4,2,3,1] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 4
[7,8,6,5,3,2,4,1] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 4
[7,8,5,4,3,2,6,1] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 4
[6,7,5,4,3,2,8,1] => [2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 4
[7,8,6,5,4,3,1,2] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 4
[7,8,5,6,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[6,7,5,8,3,4,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[5,6,7,8,3,4,1,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[7,8,4,5,3,6,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[7,8,3,4,5,6,1,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 0
[6,7,4,5,3,8,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[5,6,4,7,3,8,1,2] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[4,5,6,7,3,8,1,2] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[5,6,3,4,7,8,1,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 0
[4,5,3,6,7,8,1,2] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 0
[7,8,6,5,4,2,1,3] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 4
[7,8,6,5,4,1,2,3] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[7,8,6,5,3,2,1,4] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 4
[7,8,5,6,2,3,1,4] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[6,7,5,8,2,3,1,4] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[5,6,7,8,2,3,1,4] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[7,8,6,5,3,1,2,4] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[7,8,6,5,2,1,3,4] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[7,8,5,6,1,2,3,4] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[6,7,5,8,1,2,3,4] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[5,6,7,8,1,2,3,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[7,8,6,4,3,2,1,5] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 4
[7,8,6,4,2,1,3,5] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[7,8,6,3,2,1,4,5] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[8,7,6,1,2,3,4,5] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[7,8,5,4,3,2,1,6] => [2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 4
[7,8,4,5,2,3,1,6] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[7,8,3,4,2,5,1,6] => [2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 0
[7,8,2,3,4,5,1,6] => [2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 0
[7,8,5,4,3,1,2,6] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[7,8,5,4,2,1,3,6] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[7,8,4,5,1,2,3,6] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[7,8,5,3,2,1,4,6] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[8,7,5,1,2,3,4,6] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[7,8,4,3,2,1,5,6] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St001067
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 70%
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [[.,.],.]
=> [1,1,0,0]
=> 0
[2,1] => [[.,.],.]
=> [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[8,7,6,5,4,3,2,1] => [[[[[[[[.,.],.],.],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[7,8,6,5,4,3,2,1] => [[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[8,6,7,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 5
[7,6,8,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 5
[6,7,8,5,4,3,2,1] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5
[8,7,5,6,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 5
[8,6,5,7,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 5
[7,6,5,8,4,3,2,1] => [[[[[[[.,.],.],[.,.]],.],.],.],.]
=> [.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 5
[5,6,7,8,4,3,2,1] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[8,7,6,4,5,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5
[8,7,5,4,6,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5
[7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 5
[4,5,6,7,8,3,2,1] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[7,8,6,5,3,4,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[7,8,6,4,3,5,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[7,8,5,4,3,6,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[6,7,5,4,3,8,2,1] => [[[[[[.,[.,.]],.],.],[.,.]],.],.]
=> [.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[3,4,5,6,7,8,2,1] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[7,8,6,5,4,2,3,1] => [[[[[[.,[.,.]],.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[7,8,6,5,3,2,4,1] => [[[[[[.,[.,.]],.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[7,8,5,4,3,2,6,1] => [[[[[[.,[.,.]],.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[7,8,2,3,4,5,6,1] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[6,7,5,4,3,2,8,1] => [[[[[[.,[.,.]],.],.],.],[.,.]],.]
=> [.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[6,7,2,3,4,5,8,1] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[4,5,2,3,6,7,8,1] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[3,4,2,5,6,7,8,1] => [[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[2,3,4,5,6,7,8,1] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[8,7,6,5,4,3,1,2] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[7,8,6,5,4,3,1,2] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[7,8,5,6,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[6,7,5,8,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[5,6,7,8,3,4,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[7,8,4,5,3,6,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[7,8,3,4,5,6,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
[6,7,4,5,3,8,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[5,6,4,7,3,8,1,2] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[4,5,6,7,3,8,1,2] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[5,6,3,4,7,8,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
[4,5,3,6,7,8,1,2] => [[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]
=> [[.,.],[[[[.,.],.],.],[[.,.],.]]]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
[3,4,5,6,7,8,1,2] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[8,7,6,5,4,2,1,3] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[7,8,6,5,4,2,1,3] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[8,7,6,5,4,1,2,3] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[7,8,6,5,4,1,2,3] => [[[[[.,[.,.]],.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[8,7,6,5,3,2,1,4] => [[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[7,8,6,5,3,2,1,4] => [[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 4
[7,8,5,6,2,3,1,4] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[6,7,5,8,2,3,1,4] => [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 0
[5,6,7,8,2,3,1,4] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[8,7,6,5,3,1,2,4] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001640
(load all 6 compositions to match this statistic)
(load all 6 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
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 70%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 70%
Values
[1] => [.,.]
=> [1,0]
=> [1] => 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [2,1] => 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [1,2] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,3,2] => 0
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 0
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 0
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 0
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ? = 0
[1,3,2,7,6,5,4] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 3
[1,4,2,7,6,5,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 3
[1,4,3,2,6,5,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,4,3,2,7,5,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,4,3,2,7,6,5] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 3
[1,5,2,7,6,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 3
[1,5,3,2,6,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,5,3,2,7,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,5,3,2,7,6,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 3
[1,5,4,2,6,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,5,4,2,7,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,5,4,2,7,6,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 3
[1,5,4,3,2,6,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 2
[1,5,4,3,2,7,6] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 3
[1,5,4,3,6,2,7] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => ? = 1
[1,5,4,3,7,2,6] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => ? = 1
[1,5,4,3,7,6,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,1,3,6,4,5,7] => ? = 3
[1,6,2,7,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 3
[1,6,3,2,5,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,6,3,2,7,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,6,3,2,7,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 3
[1,6,4,2,5,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,6,4,2,7,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,6,4,2,7,5,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 3
[1,6,4,3,2,5,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 2
[1,6,4,3,2,7,5] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 3
[1,6,4,3,5,2,7] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => ? = 1
[1,6,4,3,7,2,5] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => ? = 1
[1,6,4,3,7,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,1,3,6,4,5,7] => ? = 3
[1,6,5,2,4,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,6,5,2,7,3,4] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,6,5,2,7,4,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 3
[1,6,5,3,2,4,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 2
[1,6,5,3,2,7,4] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 3
[1,6,5,3,4,2,7] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => ? = 1
[1,6,5,3,7,2,4] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => ? = 1
[1,6,5,3,7,4,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [2,1,3,6,4,5,7] => ? = 3
[1,6,5,4,2,3,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 2
[1,6,5,4,2,7,3] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 3
[1,6,5,4,3,2,7] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 3
[1,6,5,4,3,7,2] => [.,[[[[[.,.],.],.],[.,.]],.]]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [2,1,3,4,7,5,6] => ? = 3
[1,6,5,4,7,2,3] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,1,3,6,4,7,5] => ? = 1
[1,6,5,4,7,3,2] => [.,[[[[[.,.],.],[.,.]],.],.]]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,1,3,7,4,5,6] => ? = 3
[1,6,5,7,4,3,2] => [.,[[[[[.,.],[.,.]],.],.],.]]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 3
[1,7,2,6,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 3
[1,7,3,2,5,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,7,3,2,6,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
[1,7,3,2,6,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 3
[1,7,4,2,5,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000237
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> [1] => 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> [1,2] => 0
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> [2,1] => 1
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[3,1,2] => [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 0
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 0
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 0
[1,3,2,7,6,5,4] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => ? = 3
[1,4,2,7,6,5,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => ? = 3
[1,4,3,2,6,5,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,4,3,2,7,5,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,4,3,2,7,6,5] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3
[1,5,2,7,6,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => ? = 3
[1,5,3,2,6,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,5,3,2,7,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,5,3,2,7,6,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3
[1,5,4,2,6,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,5,4,2,7,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,5,4,2,7,6,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3
[1,5,4,3,2,6,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => ? = 2
[1,5,4,3,2,7,6] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,6,4,5,3,7,2] => ? = 3
[1,5,4,3,7,6,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => ? = 3
[1,6,2,7,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => ? = 3
[1,6,3,2,5,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,6,3,2,7,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,6,3,2,7,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3
[1,6,4,2,5,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,6,4,2,7,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,6,4,2,7,5,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3
[1,6,4,3,2,5,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => ? = 2
[1,6,4,3,2,7,5] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,6,4,5,3,7,2] => ? = 3
[1,6,4,3,7,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => ? = 3
[1,6,5,2,4,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,6,5,2,7,3,4] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,6,5,2,7,4,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3
[1,6,5,3,2,4,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => ? = 2
[1,6,5,3,2,7,4] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,6,4,5,3,7,2] => ? = 3
[1,6,5,3,7,4,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => ? = 3
[1,6,5,4,2,3,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => ? = 2
[1,6,5,4,2,7,3] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,6,4,5,3,7,2] => ? = 3
[1,6,5,4,3,7,2] => [.,[[[[[.,.],.],.],[.,.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => ? = 3
[1,6,5,4,7,3,2] => [.,[[[[[.,.],.],[.,.]],.],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => ? = 3
[1,6,5,7,4,3,2] => [.,[[[[[.,.],[.,.]],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,6,2] => ? = 3
[1,7,2,6,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => ? = 3
[1,7,3,2,5,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,7,3,2,6,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,7,3,2,6,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3
[1,7,4,2,5,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,7,4,2,6,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,7,4,2,6,5,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3
[1,7,4,3,2,5,6] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => ? = 2
[1,7,4,3,2,6,5] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,6,4,5,3,7,2] => ? = 3
[1,7,4,3,6,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => ? = 3
[1,7,5,2,4,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,7,5,2,6,3,4] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 1
[1,7,5,2,6,4,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 3
[1,7,5,3,2,4,6] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,6,4,5,3,2,7] => ? = 2
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St001172
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 60%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001172: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1,0]
=> [1,1,0,0]
=> 0
[1,2] => [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 0
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,2,3] => [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,3,1] => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[1,2,3,4,5,6,7] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,3,2,7,6,5,4] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,4,2,7,6,5,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,4,3,2,6,5,7] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,4,3,2,7,5,6] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,4,3,2,7,6,5] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,5,2,7,6,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,5,3,2,6,4,7] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,5,3,2,7,4,6] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,5,3,2,7,6,4] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,5,4,2,6,3,7] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,5,4,2,7,3,6] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,5,4,2,7,6,3] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,5,4,3,2,6,7] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,5,4,3,2,7,6] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3
[1,5,4,3,6,2,7] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,5,4,3,7,2,6] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,5,4,3,7,6,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,2,7,5,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,6,3,2,5,4,7] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,6,3,2,7,4,5] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,6,3,2,7,5,4] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,4,2,5,3,7] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,6,4,2,7,3,5] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,6,4,2,7,5,3] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,4,3,2,5,7] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,6,4,3,2,7,5] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3
[1,6,4,3,5,2,7] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,6,4,3,7,2,5] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,6,4,3,7,5,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,5,2,4,3,7] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,6,5,2,7,3,4] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,6,5,2,7,4,3] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,5,3,2,4,7] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,6,5,3,2,7,4] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3
[1,6,5,3,4,2,7] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,6,5,3,7,2,4] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,6,5,3,7,4,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,5,4,2,3,7] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[1,6,5,4,2,7,3] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3
[1,6,5,4,3,2,7] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 3
[1,6,5,4,3,7,2] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3
[1,6,5,4,7,2,3] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,6,5,4,7,3,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,5,7,4,3,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,7,2,6,5,4,3] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,7,3,2,5,4,6] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,7,3,2,6,4,5] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
[1,7,3,2,6,5,4] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,7,4,2,5,3,6] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 1
Description
The number of 1-rises at odd height of a Dyck path.
Matching statistic: St001223
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 60%
Mp00016: Binary trees —left-right symmetry⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 60%
Values
[1] => [.,.]
=> [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [[.,.],.]
=> [1,1,0,0]
=> 0
[2,1] => [[.,.],.]
=> [.,[.,.]]
=> [1,0,1,0]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [.,[[.,.],.]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 0
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> 0
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,2,3,4,5,6,7] => [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,3,2,7,6,5,4] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3
[1,4,2,7,6,5,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3
[1,4,3,2,6,5,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,4,3,2,7,5,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,4,3,2,7,6,5] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,5,2,7,6,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3
[1,5,3,2,6,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,5,3,2,7,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,5,3,2,7,6,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,5,4,2,6,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,5,4,2,7,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,5,4,2,7,6,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,5,4,3,2,6,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,5,4,3,2,7,6] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,5,4,3,6,2,7] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,5,4,3,7,2,6] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,5,4,3,7,6,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,2,7,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3
[1,6,3,2,5,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,3,2,7,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,3,2,7,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,4,2,5,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,4,2,7,3,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,4,2,7,5,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,4,3,2,5,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,6,4,3,2,7,5] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,6,4,3,5,2,7] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,4,3,7,2,5] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,4,3,7,5,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,5,2,4,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,5,2,7,3,4] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,5,2,7,4,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,5,3,2,4,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,6,5,3,2,7,4] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,6,5,3,4,2,7] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,5,3,7,2,4] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,5,3,7,4,2] => [.,[[[[.,.],.],[[.,.],.]],.]]
=> [[.,[[.,[.,.]],[.,[.,.]]]],.]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,5,4,2,3,7] => [.,[[[[.,.],.],.],[.,[.,.]]]]
=> [[[[.,.],.],[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[1,6,5,4,2,7,3] => [.,[[[[.,.],.],.],[[.,.],.]]]
=> [[[.,[.,.]],[.,[.,[.,.]]]],.]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,6,5,4,3,2,7] => [.,[[[[[.,.],.],.],.],[.,.]]]
=> [[[.,.],[.,[.,[.,[.,.]]]]],.]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,6,5,4,3,7,2] => [.,[[[[[.,.],.],.],[.,.]],.]]
=> [[.,[[.,.],[.,[.,[.,.]]]]],.]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 3
[1,6,5,4,7,2,3] => [.,[[[[.,.],.],[.,.]],[.,.]]]
=> [[[.,.],[[.,.],[.,[.,.]]]],.]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,6,5,4,7,3,2] => [.,[[[[[.,.],.],[.,.]],.],.]]
=> [[.,[.,[[.,.],[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,6,5,7,4,3,2] => [.,[[[[[.,.],[.,.]],.],.],.]]
=> [[.,[.,[.,[[.,.],[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3
[1,7,2,6,5,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> [[[.,[.,[.,[.,.]]]],[.,.]],.]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 3
[1,7,3,2,5,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,7,3,2,6,4,5] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
[1,7,3,2,6,5,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> [[[.,[.,[.,.]]],[.,[.,.]]],.]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 3
[1,7,4,2,5,3,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[[[.,.],[.,.]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> ? = 1
Description
Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!