- FindStat

Your data matches 51 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000372
(load all 4 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000372: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => [1,2] => 0
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [2,1] => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [1,3,2] => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [3,1,2] => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => 0
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => [1,2,3] => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,4,3,2] => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [4,1,3,2] => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,3,1] => 0
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,4,2,3] => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,2,3,4] => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [4,1,2,3] => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,4,2,1] => 0
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,3,4,1] => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,3,2,4] => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,4,5,3,2] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [1,5,3,4,2] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [1,3,4,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [5,1,3,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,4,5,1,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [1,4,3,5,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [2,4,5,3,1] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [1,5,4,2,3] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [5,1,4,2,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [1,5,2,3,4] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => [1,2,4,3,5] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [5,1,2,3,4] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [4,5,1,2,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [1,4,5,2,3] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => [1,4,2,3,5] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [5,4,1,2,3] => 1

Description

The number of mid points of increasing subsequences of length 3 in a permutation. For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is the number of indices

$j$ such that there exist indices

$i,k$ with

$i < j < k$ and

$\pi(i) < \pi(j) < \pi(k)$ . The generating function is given by [1].

Matching statistic: St001082
(load all 6 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001082: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1] => [1] => ? = 0
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => [1,2] => 0
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [2,1] => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [1,3,2] => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [3,1,2] => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [2,3,1] => 0
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => [1,2,3] => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,4,3,2] => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [4,1,3,2] => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,4,1,2] => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [4,3,1,2] => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,3,1] => 0
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,4,2,3] => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,2,3,4] => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [4,1,2,3] => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [3,4,2,1] => 0
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,3,4,1] => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [1,2,4,3] => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,5,4,3,2] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [5,1,4,3,2] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [4,5,1,3,2] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,4,5,3,2] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [5,4,1,3,2] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,5,4,1,2] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [1,5,3,4,2] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [1,3,4,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [5,1,3,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [3,4,5,1,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => [1,3,5,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [5,4,3,1,2] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [2,5,4,3,1] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [5,2,4,3,1] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [2,4,5,3,1] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [1,5,4,2,3] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [5,1,4,2,3] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [1,5,2,3,4] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => [1,2,3,5,4] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [5,1,2,3,4] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [4,5,1,2,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [1,4,5,2,3] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => [1,2,5,3,4] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [5,4,1,2,3] => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,5,4,2,1] => 0

Description

The number of boxed occurrences of 123 in a permutation. This is the number of occurrences of the pattern

$123$ 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: St001682
(load all 7 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001682: Permutations ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [1,1,0,0]
 => [1,2] => [1,2] => 0
[1,1,0,0]
 => [1,0,1,0]
 => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [2,3,1] => [1,3,2] => 0
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,2,4,1] => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [3,4,2,1] => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [2,1,4,3] => 0
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [2,4,3,1] => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,4,3,2] => 0
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [4,3,2,5,1] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [4,3,1,5,2] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,4,3,2,5] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [4,3,5,2,1] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [4,2,1,5,3] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [4,2,5,3,1] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [2,4,3,1,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,2,4,3,5] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [2,4,3,5,1] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [4,1,5,3,2] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,4,2,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [2,1,4,3,5] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [4,5,3,2,1] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [3,2,1,5,4] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [3,2,5,4,1] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [3,1,5,4,2] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,3,2,5,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [3,5,4,2,1] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [3,4,2,1,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [3,4,2,5,1] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [2,3,4,1,5] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [2,3,4,5,1] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [3,4,1,5,2] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [3,1,4,2,5] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [2,3,1,4,5] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [3,4,5,2,1] => 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,1,7] => [3,2,1,4,6,5,7] => ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,6,1,7] => [2,1,4,3,6,5,7] => ? = 4
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,1,7] => [2,3,1,4,6,5,7] => ? = 4
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,1,7] => [2,1,3,4,6,5,7] => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 5
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [4,3,5,1,2,6,7] => [4,2,1,5,3,6,7] => ? = 5
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [4,2,3,1,5,6,7] => [2,4,3,1,5,6,7] => ? = 5
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [3,2,4,1,5,6,7] => [2,1,4,3,5,6,7] => ? = 5
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,1,6,7] => [3,2,1,5,4,6,7] => ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1,2,5,6,7] => [3,4,2,1,5,6,7] => ? = 5
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => ? = 5
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 5
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [3,4,1,2,5,6,7] => [3,1,4,2,5,6,7] => ? = 5
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => ? = 5
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,1,6,7] => [2,1,5,4,3,6,7] => ? = 5
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,1,6,7] => [2,3,1,5,4,6,7] => ? = 5
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1,3,5,6,7] => [3,2,4,1,5,6,7] => ? = 5
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,1,6,7] => [2,1,3,5,4,6,7] => ? = 5
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 5

Description

The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation.

Matching statistic: St000371
(load all 8 compositions to match this statistic)

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000371: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1,0]
 => [1] => 0
[1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,2] => 0
[1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 0
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 2
[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]
 => [5,4,3,2,1] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [7,5,4,3,6,2,1] => ? = 5
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [7,3,2,6,5,4,1] => ? = 5
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [7,4,3,6,5,2,1] => ? = 5
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [7,6,4,3,5,2,1] => ? = 5
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [3,2,7,5,6,4,1] => ? = 4
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [7,3,2,5,6,4,1] => ? = 5
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [7,4,3,5,6,2,1] => ? = 5
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [7,2,6,5,4,3,1] => ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [7,3,6,5,4,2,1] => ? = 5
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [7,6,3,5,4,2,1] => ? = 5
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [7,6,4,5,3,2,1] => ? = 5
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [7,2,5,4,6,3,1] => ? = 5
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [7,3,5,4,6,2,1] => ? = 5
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [7,5,3,4,6,2,1] => ? = 5
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [7,2,3,6,5,4,1] => ? = 5
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [7,2,4,6,5,3,1] => ? = 5
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [7,2,6,4,5,3,1] => ? = 5
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [7,3,4,6,5,2,1] => ? = 5
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [7,3,6,4,5,2,1] => ? = 5
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [7,6,3,4,5,2,1] => ? = 5
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [7,2,3,5,6,4,1] => ? = 5
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [7,2,4,5,6,3,1] => ? = 5
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [7,3,4,5,6,2,1] => ? = 5

Description

The number of mid points of decreasing subsequences of length 3 in a permutation. For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is the number of indices

$j$ such that there exist indices

$i,k$ with

$i < j < k$ and

$\pi(i) > \pi(j) > \pi(k)$ . In other words, this is the number of indices that are neither left-to-right maxima nor right-to-left minima. This statistic can also be expressed as the number of occurrences of the mesh pattern ([3,2,1], {(0,2),(0,3),(2,0),(3,0)}): the shading fixes the first and the last element of the decreasing subsequence. See also [[St000119]].

Matching statistic: St001229
(load all 22 compositions to match this statistic)

Mapping

St001229: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => 0
[1,0,1,0]
 => 0
[1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 4
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 4
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 4
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 4
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 4
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 5
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 5
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 5
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 5
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 5
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 4
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 5
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 5
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => ? = 5
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => ? = 5
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 5
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => ? = 5
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => ? = 5
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => ? = 5
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => ? = 5
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => ? = 5
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => ? = 5
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => ? = 5
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => ? = 5
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 5
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => ? = 5
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => ? = 5
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 5

Description

The vector space dimension of the first extension group between the Jacobson radical J and J^2. The vector space dimension of

$Ext_A^1(J,J^2)$ .

Matching statistic: St000316

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => [1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [1,1,0,0]
 => [2,1] => [1,2] => 0
[1,1,0,0]
 => [1,0,1,0]
 => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2,1,3] => [1,2,3] => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,2,3] => 0
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => [1,3,2] => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [1,2,4,3] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [1,3,4,2] => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [1,4,2,3] => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [1,3,2,4] => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [1,2,4,5,3] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [1,2,5,3,4] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [1,2,4,3,5] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [1,2,5,3,4] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [1,2,3,5,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => [1,3,4,5,2] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => [1,3,4,2,5] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => [1,4,5,2,3] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,3,4,2,1] => [1,5,2,3,4] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => [1,4,2,3,5] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => [1,3,5,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5,3,2,4,1] => [1,5,2,3,4] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => [1,3,2,4,5] => 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,7,6,5,4,1] => [1,2,3,7,4,6,5] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [2,7,4,5,6,3,1] => [1,2,7,3,4,5,6] => ? = 4
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [2,7,4,6,5,3,1] => [1,2,7,3,4,6,5] => ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [2,7,3,5,6,4,1] => [1,2,7,3,4,5,6] => ? = 4
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [2,7,5,4,6,3,1] => [1,2,7,3,5,6,4] => ? = 4
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,7,6,5,4,3,1] => [1,2,7,3,6,4,5] => ? = 4
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [2,7,3,6,5,4,1] => [1,2,7,3,4,6,5] => ? = 4
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [2,7,6,4,5,3,1] => [1,2,7,3,6,4,5] => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [7,3,4,5,6,2,1] => [1,7,2,3,4,5,6] => ? = 5
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [7,3,4,6,5,2,1] => [1,7,2,3,4,6,5] => ? = 5
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [7,3,2,5,6,4,1] => [1,7,2,3,4,5,6] => ? = 5
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [7,3,5,4,6,2,1] => [1,7,2,3,5,6,4] => ? = 5
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [7,3,6,5,4,2,1] => [1,7,2,3,6,4,5] => ? = 5
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,2,7,6,5,4,1] => [1,3,7,2,4,6,5] => ? = 4
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [7,3,2,6,5,4,1] => [1,7,2,3,4,6,5] => ? = 5
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [7,3,6,4,5,2,1] => [1,7,2,3,6,4,5] => ? = 5
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [7,2,4,5,6,3,1] => [1,7,2,3,4,5,6] => ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [7,4,3,5,6,2,1] => [1,7,2,4,5,6,3] => ? = 5
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [7,5,4,3,6,2,1] => [1,7,2,5,6,3,4] => ? = 5
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [7,6,4,5,3,2,1] => [1,7,2,6,3,4,5] => ? = 5
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [7,2,4,6,5,3,1] => [1,7,2,3,4,6,5] => ? = 5
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [7,4,3,6,5,2,1] => [1,7,2,4,6,3,5] => ? = 5
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [7,6,4,3,5,2,1] => [1,7,2,6,3,4,5] => ? = 5
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [7,2,3,5,6,4,1] => [1,7,2,3,4,5,6] => ? = 5
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [7,2,5,4,6,3,1] => [1,7,2,3,5,6,4] => ? = 5
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [7,2,6,5,4,3,1] => [1,7,2,3,6,4,5] => ? = 5
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [7,5,3,4,6,2,1] => [1,7,2,5,6,3,4] => ? = 5
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [7,6,3,5,4,2,1] => [1,7,2,6,3,4,5] => ? = 5
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [7,6,5,4,3,2,1] => [1,7,2,6,3,5,4] => ? = 5
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [7,2,3,6,5,4,1] => [1,7,2,3,4,6,5] => ? = 5
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [7,2,6,4,5,3,1] => [1,7,2,3,6,4,5] => ? = 5
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [7,6,3,4,5,2,1] => [1,7,2,6,3,4,5] => ? = 5

Description

The number of non-left-to-right-maxima of a permutation. An integer

$\sigma_i$ in the one-line notation of a permutation

$\sigma$ is a **non-left-to-right-maximum** if there exists a

$j < i$ such that

$\sigma_j > \sigma_i$ .

Matching statistic: St001687
(load all 2 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001687: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 86%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => [1] => [1] => [1] => 0
[1,0,1,0]
 => [2,1] => [2,1] => [1,2] => 0
[1,1,0,0]
 => [1,2] => [1,2] => [2,1] => 0
[1,0,1,0,1,0]
 => [2,3,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => [3,1,2] => 0
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => [2,3,1] => 0
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => [2,1,3] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 0
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,2,1,3] => [3,1,2,4] => 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 0
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 0
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,3,1,2] => [2,1,3,4] => 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [4,1,3,2] => [2,3,1,4] => 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,2,4] => [4,2,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => [3,2,1,4] => 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => [4,5,1,2,3] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5,3,2,1,4] => [4,1,2,3,5] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [5,4,2,1,3] => [3,1,2,4,5] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [5,2,1,4,3] => [3,4,1,2,5] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [4,2,1,3,5] => [5,3,1,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,5,3,4] => [4,3,5,1,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5,2,1,3,4] => [4,3,1,2,5] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,5,3,2,4] => [4,2,3,5,1] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [5,4,3,1,2] => [2,1,3,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [4,3,1,2,5] => [5,2,1,3,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [5,4,1,3,2] => [2,3,1,4,5] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5,1,4,3,2] => [2,3,4,1,5] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,1,3,2,5] => [5,2,3,1,4] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,2,5,4] => [4,5,2,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [5,3,1,2,4] => [4,2,1,3,5] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [5,1,3,2,4] => [4,2,3,1,5] => 3
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,3,6,7,1,4,5] => [7,3,2,1,6,4,5] => [5,4,6,1,2,3,7] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,7,1,3] => [7,2,1,6,5,4,3] => [3,4,5,6,1,2,7] => ? = 4
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,6,7,1,3,5] => [7,2,1,6,4,3,5] => [5,3,4,6,1,2,7] => ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [2,5,1,6,7,3,4] => [7,2,1,3,6,5,4] => [4,5,6,3,1,2,7] => ? = 4
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [2,5,6,1,7,3,4] => [7,2,1,6,5,3,4] => [4,3,5,6,1,2,7] => ? = 4
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [2,5,6,7,1,3,4] => [7,2,1,6,3,5,4] => [4,5,3,6,1,2,7] => ? = 4
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,6,1,7,3,4,5] => [7,2,1,3,6,4,5] => [5,4,6,3,1,2,7] => ? = 4
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,6,7,1,3,4,5] => [7,2,1,6,3,4,5] => [5,4,3,6,1,2,7] => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,7,1,2] => [7,1,6,5,4,3,2] => [2,3,4,5,6,1,7] => ? = 5
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [3,4,6,7,1,2,5] => [7,1,6,4,3,2,5] => [5,2,3,4,6,1,7] => ? = 5
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => [3,5,1,6,7,2,4] => [7,1,3,2,6,5,4] => [4,5,6,2,3,1,7] => ? = 5
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [3,5,6,1,7,2,4] => [7,1,6,5,3,2,4] => [4,2,3,5,6,1,7] => ? = 5
[1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [3,5,6,7,1,2,4] => [7,1,6,3,2,5,4] => [4,5,2,3,6,1,7] => ? = 5
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [3,1,6,7,2,4,5] => [7,3,1,2,6,4,5] => [5,4,6,2,1,3,7] => ? = 4
[1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [3,6,1,7,2,4,5] => [7,1,3,2,6,4,5] => [5,4,6,2,3,1,7] => ? = 5
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [3,6,7,1,2,4,5] => [7,1,6,3,2,4,5] => [5,4,2,3,6,1,7] => ? = 5
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [4,1,5,6,7,2,3] => [7,1,2,6,5,4,3] => [3,4,5,6,2,1,7] => ? = 5
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [4,5,1,6,7,2,3] => [7,1,6,5,4,2,3] => [3,2,4,5,6,1,7] => ? = 5
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [4,5,6,1,7,2,3] => [7,1,6,5,2,4,3] => [3,4,2,5,6,1,7] => ? = 5
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [4,5,6,7,1,2,3] => [7,1,6,2,5,4,3] => [3,4,5,2,6,1,7] => ? = 5
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [4,1,6,7,2,3,5] => [7,1,2,6,4,3,5] => [5,3,4,6,2,1,7] => ? = 5
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [4,6,1,7,2,3,5] => [7,1,6,4,2,3,5] => [5,3,2,4,6,1,7] => ? = 5
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [4,6,7,1,2,3,5] => [7,1,6,2,4,3,5] => [5,3,4,2,6,1,7] => ? = 5
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [5,1,2,6,7,3,4] => [7,1,2,3,6,5,4] => [4,5,6,3,2,1,7] => ? = 5
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [5,1,6,2,7,3,4] => [7,1,2,6,5,3,4] => [4,3,5,6,2,1,7] => ? = 5
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [5,1,6,7,2,3,4] => [7,1,2,6,3,5,4] => [4,5,3,6,2,1,7] => ? = 5
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [5,6,1,2,7,3,4] => [7,1,6,5,2,3,4] => [4,3,2,5,6,1,7] => ? = 5
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [5,6,1,7,2,3,4] => [7,1,6,2,3,5,4] => [4,5,3,2,6,1,7] => ? = 5
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [5,6,7,1,2,3,4] => [7,1,6,2,5,3,4] => [4,3,5,2,6,1,7] => ? = 5
[1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [6,1,2,7,3,4,5] => [7,1,2,3,6,4,5] => [5,4,6,3,2,1,7] => ? = 5
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [6,1,7,2,3,4,5] => [7,1,2,6,3,4,5] => [5,4,3,6,2,1,7] => ? = 5
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [6,7,1,2,3,4,5] => [7,1,6,2,3,4,5] => [5,4,3,2,6,1,7] => ? = 5

Description

The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation.

Matching statistic: St001596
(load all 6 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => [1,0]
 => [[1],[]]
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => [[2],[]]
 => 0
[1,1,0,0]
 => [1,0,1,0]
 => [[1,1],[]]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => 0
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => 0
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => ? = 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => ? = 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => ? = 3
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [[5,5],[2]]
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [[4,4,4],[2,2]]
 => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [[5,4],[1]]
 => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [[5,5],[1]]
 => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[4,4,4],[3,1]]
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [[4,4,3],[2,1]]
 => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [[4,4,4],[2,1]]
 => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[4,4,2],[1,1]]
 => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [[4,3,3],[1,1]]
 => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [[4,4,3],[1,1]]
 => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1,1]]
 => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [[3,3,3,2],[1,1,1]]
 => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [[3,3,3,3],[1,1,1]]
 => ? = 3
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [[4,4,1],[1]]
 => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3,1],[1,1]]
 => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [[4,4,2],[2]]
 => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [[5,3],[]]
 => ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[4,4,3],[3]]
 => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [[5,4],[]]
 => ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [[5,5],[]]
 => ? = 4
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[4,3,3],[2]]
 => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [[4,4,3],[2]]
 => ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [[4,4,4],[2]]
 => ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [[3,3,3,3],[2,2]]
 => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [[4,3,2],[1]]
 => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [[4,4,2],[1]]
 => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1]]
 => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [[4,3,3],[1]]
 => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [[4,4,3],[1]]
 => ? = 4
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [[4,4,4],[1]]
 => ? = 4
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1]]
 => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1]]
 => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [[3,3,3,2],[1,1]]
 => ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [[3,3,3,3],[1,1]]
 => ? = 4
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [[4,3,1],[]]
 => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => ? = 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3,1],[2]]
 => ? = 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2,1],[1]]
 => ? = 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3,1],[1]]
 => ? = 3
[1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[4,2,2],[]]
 => ? = 2
[1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [[3,3,2,2],[2]]
 => ? = 2
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[4,3,2],[]]
 => ? = 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [[4,4,2],[]]
 => ? = 4

Description

The number of two-by-two squares inside a skew partition. This is, the number of cells

$(i,j)$ in a skew partition for which the box

$(i+1,j+1)$ is also a cell inside the skew partition.

Matching statistic: St001633
(load all 4 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001633: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => [1,0]
 => ([],1)
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => ([(0,1)],2)
 => 0
[1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 0
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 3
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 3
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 4
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 4
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 4
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 4
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 3
[1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 2
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 4

Description

The number of simple modules with projective dimension two in the incidence algebra of the poset.

Matching statistic: St000362

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 54%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => [1,0]
 => ([],1)
 => ([],1)
 => 0
[1,0,1,0]
 => [1,1,0,0]
 => ([(0,1)],2)
 => ([],2)
 => 0
[1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(3,4)],5)
 => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(3,4)],5)
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ([(4,5)],6)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ([(3,6),(4,5),(5,6)],7)
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ([(3,6),(4,5)],7)
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 3
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ([(4,5)],6)
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 0
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ([(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? = 3
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ([(4,7),(5,6)],8)
 => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
 => ? = 3
[1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ([(4,7),(5,6)],8)
 => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 4
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ([(3,4),(5,8),(6,7),(7,8)],9)
 => ? = 3
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
 => ? = 4
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ([(4,7),(5,6)],8)
 => ? = 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ([(3,4),(5,8),(6,7),(7,8)],9)
 => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ([(4,7),(5,6)],8)
 => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? = 3
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ([(2,9),(3,8),(4,6),(5,7),(6,8),(7,9),(8,9)],10)
 => ? = 4
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ([(2,8),(3,4),(3,10),(4,9),(5,9),(5,10),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 4
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? = 3
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ([(4,7),(5,6)],8)
 => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ([(3,4),(5,8),(6,7),(7,8)],9)
 => ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
 => ? = 4
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ([(4,7),(5,6)],8)
 => ? = 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? = 3
[1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ([(4,7),(5,6),(6,7)],8)
 => ? = 2
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
 => ? = 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ([(2,9),(3,8),(4,5),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
 => ? = 4

Description

The size of a minimal vertex cover of a graph. A '''vertex cover''' of a graph

$G$ is a subset

$S$ of the vertices of

$G$ such that each edge of

$G$ contains at least one vertex of

$S$ . Finding a minimal vertex cover is an NP-hard optimization problem.

The following 41 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000387The matching number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001725The harmonious chromatic number of a graph. St000456The monochromatic index of a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St001480The number of simple summands of the module J^2/J^3. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000028The number of stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St001330The hat guessing number of a graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000058The order of a permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001864The number of excedances of a signed permutation. St001115The number of even descents of a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000074The number of special entries. St000516The number of stretching pairs of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.