Your data matches 130 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001394
(load all 6 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => 0
[.,[.,.]]
 => [2,1] => 0
[[.,.],.]
 => [1,2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => 0
[.,[[.,.],.]]
 => [2,3,1] => 0
[[.,.],[.,.]]
 => [1,3,2] => 0
[[.,[.,.]],.]
 => [2,1,3] => 0
[[[.,.],.],.]
 => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => 0
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => 0
[[.,.],[[.,.],.]]
 => [1,3,4,2] => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => 0
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => 0
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => 0
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => 0
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => 0
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => 0
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => 0
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => 0
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => 0
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => 0
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => 0
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => 0
Description
The genus of a permutation. The genus $g(\pi)$ of a permutation $\pi\in\mathfrak S_n$ is defined via the relation $$ n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ), $$ where $\zeta = (1,2,\dots,n)$ is the long cycle and $z(\cdot)$ is the number of cycles in the permutation.
Matching statistic: St001906
(load all 8 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [2,1] => 0
[[.,.],.]
 => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => 0
[[.,.],[.,.]]
 => [1,3,2] => [2,1,3] => 0
[[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => 0
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [3,1,4,2] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,1,2] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,4,1,2] => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,1,4,3] => 0
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [2,4,1,3] => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [3,2,1,4] => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [2,3,1,4] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,3,2] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,3,4,2] => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,2,4,3] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,2,4] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,4,3,2] => 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [4,1,5,3,2] => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,1,3,2] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,5,1,3,2] => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [3,1,5,4,2] => 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [3,5,1,4,2] => 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [4,3,1,5,2] => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [3,4,1,5,2] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,3,1,2] => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [4,5,3,1,2] => 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [3,5,4,1,2] => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [4,3,5,1,2] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [3,4,5,1,2] => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [2,1,5,4,3] => 0
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [2,5,1,4,3] => 0
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [2,4,1,5,3] => 0
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [2,5,4,1,3] => 0
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [2,4,5,1,3] => 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [3,2,1,5,4] => 0
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [3,2,5,1,4] => 0
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [2,3,1,5,4] => 0
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [2,3,5,1,4] => 0
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [4,3,2,1,5] => 0
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,4,2,1,5] => 0
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [2,4,3,1,5] => 0
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [3,2,4,1,5] => 0
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [2,3,4,1,5] => 0
Description
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. Let $\pi$ be a permutation. Its total displacement [[St000830]] is $D(\pi) = \sum_i |\pi(i) - i|$, and its absolute length [[St000216]] is the minimal number $T(\pi)$ of transpositions whose product is $\pi$. Finally, let $I(\pi)$ be the number of inversions [[St000018]] of $\pi$. This statistic equals $\left(D(\pi)-T(\pi)-I(\pi)\right)/2$. Diaconis and Graham [1] proved that this statistic is always nonnegative.
Matching statistic: St000306
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
[.,[.,.]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[[.,.],.]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
Description
The bounce count of a Dyck path. For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000864
(load all 2 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [1,2] => [1,2] => 0
[[.,.],.]
 => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,2,3] => 0
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
[[.,.],[.,.]]
 => [3,1,2] => [1,2,3] => [1,2,3] => 0
[[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => [1,3,2] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 0
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 0
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 0
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The number of circled entries of the shifted recording tableau of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the number of circled entries in $Q$.
Matching statistic: St001083
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [2,1] => [1,2] => 0
[[.,.],.]
 => [1,2] => [1,2] => [2,1] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => [2,1,3] => 0
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [1,2,3] => 0
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [2,3,1] => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [3,1,2] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [3,2,1] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [2,3,4,1] => [3,2,1,4] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [2,4,3,1] => [2,1,3,4] => 0
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [3,2,4,1] => [2,3,1,4] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [2,4,1,3] => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,3,4,2] => [3,2,4,1] => 0
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => [4,2,1,3] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [2,3,4,5,1] => [4,3,2,1,5] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [2,3,5,4,1] => [3,2,1,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [2,4,3,5,1] => [3,4,2,1,5] => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [2,5,4,3,1] => [2,1,3,4,5] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [2,5,3,4,1] => [3,5,2,1,4] => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [3,2,4,5,1] => [4,2,3,1,5] => 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [3,2,5,4,1] => [2,3,1,4,5] => 0
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [4,3,2,5,1] => [2,3,4,1,5] => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [4,2,3,5,1] => [2,4,3,1,5] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,3,4,2,1] => [3,5,1,2,4] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [2,5,1,3,4] => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [2,3,5,1,4] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,5,3,1,4] => 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,3,4,5,2] => [4,3,2,5,1] => 0
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,3,5,4,2] => [3,2,4,5,1] => 0
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,4,3,5,2] => [3,4,2,5,1] => 0
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [3,5,2,4,1] => 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,4,5,3] => [4,3,5,1,2] => 0
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,4,5,3] => [4,3,5,2,1] => 0
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => 0
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [2,3,1,5,4] => [4,5,2,1,3] => 0
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [4,5,1,2,3] => 0
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 0
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0
Description
The number of boxed occurrences of 132 in a permutation. This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St001197
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
[.,[.,.]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[[.,.],.]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
Description
The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001205
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001205: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
[.,[.,.]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[[.,.],.]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
Description
The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Nakayama algebra and the relation to Dyck paths.
Matching statistic: St001212
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001212: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0
[.,[.,.]]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[[.,.],.]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
Description
The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.
Matching statistic: St001331
Mapping
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001331: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1,0]
 => [1] => ([],1)
 => 0
[.,[.,.]]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 0
[[.,.],.]
 => [1,1,0,0]
 => [1,2] => ([],2)
 => 0
[.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => 0
[.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 0
[[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => 0
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 0
[[[.,.],.],.]
 => [1,1,0,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => 0
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 0
[.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 0
[.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 0
[.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 0
[[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 0
[[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 0
[[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => 0
[[[.,.],.],[.,.]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => 0
[[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 0
[[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
[[[.,.],[.,.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 0
[[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 0
[[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => 0
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 0
[.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 0
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 0
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 0
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 0
[.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 0
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0
[.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 0
[.,[[[[.,.],.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 0
[[.,.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[[.,.],[.,[[.,.],.]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[.,.],[[.,.],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[.,.],[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 0
[[.,.],[[[.,.],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 0
[[.,[.,.]],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 0
[[.,[.,.]],[[.,.],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 0
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[[[.,.],.],[[.,.],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 0
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 0
[[.,[[.,.],.]],[.,.]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 0
[[[.,.],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([],5)
 => 0
[[[.,[.,.]],.],[.,.]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 0
[[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => 0
Description
The size of the minimal feedback vertex set. A feedback vertex set is a set of vertices whose removal results in an acyclic graph.
Matching statistic: St001513
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
 => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => [3,2,1] => 0
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [2,3,1] => 0
[[.,.],[.,.]]
 => [3,1,2] => [3,1,2] => [3,1,2] => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [2,3,4,1] => [4,2,3,1] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [2,4,3,1] => [3,2,4,1] => 0
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [3,4,2,1] => [4,3,1,2] => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [3,4,1,2] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 0
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [3,1,4,2] => [4,1,3,2] => 0
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,1,3,2] => [3,1,4,2] => 0
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 0
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => [3,2,1,4] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [2,3,4,5,1] => [5,2,3,4,1] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [2,3,5,4,1] => [4,2,3,5,1] => 0
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [2,4,5,3,1] => [5,2,4,1,3] => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [2,5,4,3,1] => [4,2,5,1,3] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [2,5,3,4,1] => [3,2,4,5,1] => 0
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [3,4,2,5,1] => [5,3,1,4,2] => 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [3,5,2,4,1] => [4,3,1,5,2] => 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [4,3,5,2,1] => [4,5,3,1,2] => 0
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [4,5,2,3,1] => [5,4,1,2,3] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,3,4,2,1] => [3,5,4,1,2] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [3,4,5,1,2] => 0
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [5,4,2,3,1] => [4,5,1,2,3] => 0
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [3,4,1,5,2] => 0
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [3,1,4,5,2] => [5,1,3,4,2] => 0
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [3,1,5,4,2] => [4,1,3,5,2] => 0
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [4,1,5,3,2] => [5,1,4,2,3] => 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,1,4,3,2] => [4,1,5,2,3] => 0
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [5,1,3,4,2] => [3,1,4,5,2] => 0
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [2,4,1,5,3] => [5,2,1,4,3] => 0
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [2,5,1,4,3] => [4,2,1,5,3] => 0
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [4,1,2,5,3] => [5,1,2,4,3] => 0
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,1,2,4,3] => [4,1,2,5,3] => 0
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [2,3,5,1,4] => [5,2,3,1,4] => 0
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [3,5,2,1,4] => [5,3,1,2,4] => 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [3,1,5,2,4] => [5,1,3,2,4] => 0
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [2,5,1,3,4] => [5,2,1,3,4] => 0
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [5,1,2,3,4] => [5,1,2,3,4] => 0
Description
The number of nested exceedences of a permutation. For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
The following 120 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001871The number of triconnected components of a graph. St001624The breadth of a lattice. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001621The number of atoms of a lattice. St000944The 3-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000455The second largest eigenvalue of a graph if it is integral. St000929The constant term of the character polynomial of an integer partition. St000478Another weight of a partition according to Alladi. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000996The number of exclusive left-to-right maxima of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000655The length of the minimal rise of a Dyck path. St001964The interval resolution global dimension of a poset. St001665The number of pure excedances of a permutation. St000302The determinant of the distance matrix of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000314The number of left-to-right-maxima of a permutation. St001737The number of descents of type 2 in a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001498The normalised height of a Nakayama algebra with magnitude 1. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000934The 2-degree of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001845The number of join irreducibles minus the rank of a lattice. St000741The Colin de Verdière graph invariant. St000068The number of minimal elements in a poset. St001490The number of connected components of a skew partition. St000842The breadth of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001862The number of crossings of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001890The maximum magnitude of the Möbius function of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000629The defect of a binary word. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000383The last part of an integer composition. St001867The number of alignments of type EN of a signed permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001570The minimal number of edges to add to make a graph Hamiltonian. St001811The Castelnuovo-Mumford regularity of a permutation. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000445The number of rises of length 1 of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001584The area statistic between a Dyck path and its bounce path. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices.