Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000486
(load all 6 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
 => [2,1] => 0
[2,1] => [2,1] => [[.,.],.]
 => [1,2] => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => [2,1,3] => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => [1,3,2] => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St000647
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
 => [2,1] => 0
[2,1] => [2,1] => [[.,.],.]
 => [1,2] => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => [2,1,3] => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => [1,3,2] => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
Description
The number of big descents of a permutation. For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$. The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1]. For the number of small descents, see [[St000214]].
Matching statistic: St000660
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [2,1] => [1,1,0,0]
 => 0
[2,1] => [1,2] => [1,2] => [1,0,1,0]
 => 0
[1,2,3] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[1,3,2] => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 1
[2,1,3] => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 0
[3,1,2] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 1
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 1
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 1
[1,4,3,2] => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 1
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 0
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 0
[2,4,3,1] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 0
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 0
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 0
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[4,1,3,2] => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 0
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,3,5,4,2] => [2,4,5,3,1] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,4,5,2,3] => [3,2,5,4,1] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,4,5,3,2] => [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
 => 1
Description
The number of rises of length at least 3 of a Dyck path. The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St000710
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000710: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
 => [2,1] => 0
[2,1] => [2,1] => [[.,.],.]
 => [1,2] => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => [2,1,3] => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => [1,3,2] => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
Description
The number of big deficiencies of a permutation. A big deficiency of a permutation $\pi$ is an index $i$ such that $i - \pi(i) > 1$. This statistic is equidistributed with any of the numbers of big exceedences, big descents and big ascents.
Matching statistic: St000779
(load all 2 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000779: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
 => [2,1] => 0
[2,1] => [2,1] => [[.,.],.]
 => [1,2] => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => [2,1,3] => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => [1,3,2] => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
Description
The tier of a permutation. This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$. According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as [[OEIS:A122890]] and [[OEIS:A158830]] in the form of triangles read by rows, see [sec. 4, 1].
Matching statistic: St000836
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
 => [2,1] => 0
[2,1] => [2,1] => [[.,.],.]
 => [1,2] => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => [2,1,3] => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => [1,3,2] => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
Description
The number of descents of distance 2 of a permutation. This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St001086
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => 1
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 0
[2,3,1] => [2,3,1] => 0
[3,1,2] => [3,1,2] => 0
[3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => 1
[1,2,4,3] => [1,4,3,2] => 1
[1,3,2,4] => [1,4,3,2] => 1
[1,3,4,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,4,3] => 1
[2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [2,4,1,3] => 0
[2,3,4,1] => [2,4,3,1] => 1
[2,4,1,3] => [2,4,1,3] => 0
[2,4,3,1] => [2,4,3,1] => 1
[3,1,2,4] => [3,1,4,2] => 1
[3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [3,2,4,1] => 0
[3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => [3,4,2,1] => 0
[4,1,2,3] => [4,1,3,2] => 1
[4,1,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [4,2,1,3] => 0
[4,2,3,1] => [4,2,3,1] => 0
[4,3,1,2] => [4,3,1,2] => 0
[4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,5,4,3,2] => 1
[1,2,3,5,4] => [1,5,4,3,2] => 1
[1,2,4,3,5] => [1,5,4,3,2] => 1
[1,2,4,5,3] => [1,5,4,3,2] => 1
[1,2,5,3,4] => [1,5,4,3,2] => 1
[1,2,5,4,3] => [1,5,4,3,2] => 1
[1,3,2,4,5] => [1,5,4,3,2] => 1
[1,3,2,5,4] => [1,5,4,3,2] => 1
[1,3,4,2,5] => [1,5,4,3,2] => 1
[1,3,4,5,2] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,5,4,3,2] => 1
[1,4,2,3,5] => [1,5,4,3,2] => 1
[1,4,2,5,3] => [1,5,4,3,2] => 1
[1,4,3,2,5] => [1,5,4,3,2] => 1
[1,4,3,5,2] => [1,5,4,3,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => 1
[1,4,5,3,2] => [1,5,4,3,2] => 1
[3,1,2,4,5,6,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,4,5,7,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,4,6,5,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,4,6,7,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,4,7,5,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,4,7,6,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,5,4,6,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,5,4,7,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,5,6,4,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,5,6,7,4] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,5,7,4,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,5,7,6,4] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,6,4,5,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,6,4,7,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,6,5,4,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,6,5,7,4] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,6,7,4,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,6,7,5,4] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,7,4,5,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,7,4,6,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,7,5,4,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,7,5,6,4] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,7,6,4,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,2,7,6,5,4] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,2,5,6,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,2,5,7,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,2,6,5,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,2,6,7,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,2,7,5,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,2,7,6,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,5,2,6,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,5,2,7,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,5,6,2,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,5,7,6,2] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,6,2,5,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,6,2,7,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,6,5,2,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,6,5,7,2] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,6,7,2,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,6,7,5,2] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,7,2,5,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,7,2,6,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,7,5,2,6] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,7,5,6,2] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 1
[3,1,4,7,6,5,2] => [3,1,7,6,5,4,2] => ? = 1
[3,1,5,2,4,6,7] => [3,1,7,6,5,4,2] => ? = 1
[3,1,5,2,4,7,6] => [3,1,7,6,5,4,2] => ? = 1
Description
The number of occurrences of the consecutive pattern 132 in a permutation. This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St000023
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
 => [2,1] => 0
[2,1] => [2,1] => [[.,.],.]
 => [1,2] => 0
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => [3,1,2] => 0
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => [2,1,3] => 0
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => [3,1,2] => 0
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 0
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 0
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 0
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 0
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 0
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 0
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 0
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 0
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 0
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 1
[2,1,3,4,5,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,4,5,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,4,6,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,4,6,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,4,7,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,4,7,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,5,4,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,5,4,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,5,6,4,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,5,6,7,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,5,7,4,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,5,7,6,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,6,4,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,6,4,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,6,5,4,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,6,5,7,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,6,7,4,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,6,7,5,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,7,4,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,7,4,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,7,5,4,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,7,5,6,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,7,6,4,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,3,7,6,5,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,3,5,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,3,5,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,3,6,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,3,6,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,3,7,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,3,7,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,5,3,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,5,3,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,5,6,3,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,5,6,7,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,5,7,3,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,5,7,6,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,6,3,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,6,3,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,6,5,3,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,6,5,7,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,6,7,3,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,6,7,5,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,7,3,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,7,3,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,7,5,3,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,7,5,6,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,7,6,3,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,4,7,6,5,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,5,3,4,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
[2,1,5,3,4,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1
Description
The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].
Matching statistic: St000099
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
 => [2,1] => 1 = 0 + 1
[2,1] => [2,1] => [[.,.],.]
 => [1,2] => 1 = 0 + 1
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 2 = 1 + 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => [3,1,2] => 1 = 0 + 1
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => [2,1,3] => 1 = 0 + 1
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => [3,1,2] => 1 = 0 + 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 2 = 1 + 1
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 2 = 1 + 1
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => 2 = 1 + 1
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2 = 1 + 1
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 1 = 0 + 1
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 1 = 0 + 1
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 2 = 1 + 1
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2 = 1 + 1
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 1 = 0 + 1
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => 1 = 0 + 1
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 1 = 0 + 1
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [3,4,1,2] => 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 1 = 0 + 1
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [3,1,2,4] => 1 = 0 + 1
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [4,1,2,3] => 1 = 0 + 1
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 2 = 1 + 1
[2,1,3,4,5,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,4,5,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,4,6,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,4,6,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,4,7,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,4,7,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,5,4,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,5,4,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,5,6,4,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,5,6,7,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,5,7,4,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,5,7,6,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,6,4,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,6,4,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,6,5,4,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,6,5,7,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,6,7,4,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,6,7,5,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,7,4,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,7,4,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,7,5,4,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,7,5,6,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,7,6,4,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,3,7,6,5,4] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,3,5,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,3,5,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,3,6,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,3,6,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,3,7,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,3,7,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,5,3,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,5,3,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,5,6,3,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,5,6,7,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,5,7,3,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,5,7,6,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,6,3,5,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,6,3,7,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,6,5,3,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,6,5,7,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,6,7,3,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,6,7,5,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,7,3,5,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,7,3,6,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,7,5,3,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,7,5,6,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,7,6,3,5] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,4,7,6,5,3] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,5,3,4,6,7] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
[2,1,5,3,4,7,6] => [2,1,7,6,5,4,3] => [[.,.],[[[[[.,.],.],.],.],.]]
 => [3,4,5,6,7,1,2] => ? = 1 + 1
Description
The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St001526
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001526: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => [.,[.,.]]
 => [1,1,0,0]
 => 2 = 0 + 2
[2,1] => [2,1] => [[.,.],.]
 => [1,0,1,0]
 => 2 = 0 + 2
[1,2,3] => [1,3,2] => [.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 3 = 1 + 2
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 3 = 1 + 2
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 2 = 0 + 2
[2,3,1] => [2,3,1] => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 2 = 0 + 2
[3,1,2] => [3,1,2] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 2 = 0 + 2
[3,2,1] => [3,2,1] => [[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 2 = 0 + 2
[1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 3 = 1 + 2
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 3 = 1 + 2
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => 3 = 1 + 2
[2,4,1,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => 3 = 1 + 2
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 3 = 1 + 2
[3,1,4,2] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 3 = 1 + 2
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[3,4,1,2] => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
[4,1,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 3 = 1 + 2
[4,1,3,2] => [4,1,3,2] => [[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 3 = 1 + 2
[4,2,1,3] => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[4,2,3,1] => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[4,3,1,2] => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 1 + 2
[1,2,3,4,5,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,3,4,6,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,3,5,4,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,3,5,6,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,3,6,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,3,6,5,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,4,3,5,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,4,3,6,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,4,5,3,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,4,5,6,3] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,4,6,3,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,4,6,5,3] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,5,3,4,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,5,3,6,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,5,4,3,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,5,4,6,3] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,5,6,3,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,5,6,4,3] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,6,3,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,6,3,5,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,6,4,3,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,6,4,5,3] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,6,5,3,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,2,6,5,4,3] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,2,4,5,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,2,4,6,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,2,5,4,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,2,5,6,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,2,6,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,2,6,5,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,4,2,5,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,4,2,6,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,4,5,2,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,4,5,6,2] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,4,6,2,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,4,6,5,2] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,5,2,4,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,5,2,6,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,5,4,2,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,5,4,6,2] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,5,6,2,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,5,6,4,2] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,6,2,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,6,2,5,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,6,4,2,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,6,4,5,2] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,6,5,2,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,3,6,5,4,2] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,4,2,3,5,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
[1,4,2,3,6,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1 + 2
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.