Your data matches 19 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000147
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [2]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => [3]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [2]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => [3]
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [2]
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [3,3]
 => [3]
 => 3
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [2]
 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [5,2]
 => [2]
 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,2]
 => [2]
 => 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [4,3]
 => [3]
 => 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,2]
 => [2]
 => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [4,3]
 => [3]
 => 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [5,2]
 => [2]
 => 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [4,3]
 => [3]
 => 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [5,2]
 => [2]
 => 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [5,2]
 => [2]
 => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,3]
 => [3]
 => 3
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [4,3]
 => [3]
 => 3
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [4,3]
 => [3]
 => 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [4,3]
 => [3]
 => 3
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,2]
 => [2]
 => 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,3]
 => [3]
 => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2]
 => [2]
 => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2]
 => [2]
 => 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2]
 => [2]
 => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [5,2]
 => [2]
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,2]
 => [2]
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [5,2]
 => [2]
 => 2
Description
The largest part of an integer partition.
Matching statistic: St000474
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000474: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [2]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => [3]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [2]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => [3]
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [2]
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [3,3]
 => [3]
 => 3
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [2]
 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [5,2]
 => [2]
 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,2]
 => [2]
 => 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [4,3]
 => [3]
 => 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,2]
 => [2]
 => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [4,3]
 => [3]
 => 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [5,2]
 => [2]
 => 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [4,3]
 => [3]
 => 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [5,2]
 => [2]
 => 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [5,2]
 => [2]
 => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,3]
 => [3]
 => 3
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [4,3]
 => [3]
 => 3
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [4,3]
 => [3]
 => 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [4,3]
 => [3]
 => 3
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,2]
 => [2]
 => 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,3]
 => [3]
 => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2]
 => [2]
 => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2]
 => [2]
 => 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2]
 => [2]
 => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [5,2]
 => [2]
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,2]
 => [2]
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [5,2]
 => [2]
 => 2
Description
Dyson's crank of a partition. Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as $$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$
Matching statistic: St000784
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [2]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => [3]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [2]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => [3]
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [2]
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [3,3]
 => [3]
 => 3
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [2]
 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [5,2]
 => [2]
 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,2]
 => [2]
 => 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [4,3]
 => [3]
 => 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,2]
 => [2]
 => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [4,3]
 => [3]
 => 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [5,2]
 => [2]
 => 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [4,3]
 => [3]
 => 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [5,2]
 => [2]
 => 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [4,3]
 => [3]
 => 3
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [5,2]
 => [2]
 => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [5,2]
 => [2]
 => 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,3]
 => [3]
 => 3
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [4,3]
 => [3]
 => 3
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [4,3]
 => [3]
 => 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [4,3]
 => [3]
 => 3
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,2]
 => [2]
 => 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,3]
 => [3]
 => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2]
 => [2]
 => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2]
 => [2]
 => 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2]
 => [2]
 => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [4,3]
 => [3]
 => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [5,2]
 => [2]
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,2]
 => [2]
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [4,3]
 => [3]
 => 3
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [5,2]
 => [2]
 => 2
Description
The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].
Matching statistic: St001251
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [2,2,1]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [2,2,1,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2]
 => [2,2,1,1]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => [2,2,2]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [2,2,1,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => [2,2,2]
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [2,2,1,1]
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [3,3]
 => [2,2,2]
 => 3
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [2,2,1,1]
 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [5,2]
 => [2,2,1,1,1]
 => 2
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001280
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [2,2,1]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [2,2,1,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2]
 => [2,2,1,1]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => [2,2,2]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [2,2,1,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => [2,2,2]
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [2,2,1,1]
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [3,3]
 => [2,2,2]
 => 3
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [2,2,1,1]
 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,2]
 => [2,2,1,1,1]
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [4,3]
 => [2,2,2,1]
 => 3
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [5,2]
 => [2,2,1,1,1]
 => 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000735
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000735: Standard tableaux ⟶ ℤResult quality: 60% ●values known / values provided: 75%●distinct values known / distinct values provided: 60%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [[1,2,5],[3,4]]
 => 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [[1,2,5,6],[3,4]]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2]
 => [[1,2,5,6],[3,4]]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => [[1,2,3],[4,5,6]]
 => 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [[1,2,5,6],[3,4]]
 => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => [[1,2,3],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [[1,2,5,6],[3,4]]
 => 4 = 2 + 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [3,3]
 => [[1,2,3],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [[1,2,5,6],[3,4]]
 => 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 5 = 3 + 2
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [7,1,2,8,3,4,5,9,6] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [8,1,2,5,9,3,4,6,7] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [7,1,4,9,2,3,5,6,8] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [9,1,4,8,2,3,5,6,7] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [5,9,1,2,3,8,4,6,7] => [4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => ? = 3 + 2
[1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [8,6,1,2,3,7,4,9,5] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [9,6,1,2,3,7,8,4,5] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [8,4,1,2,9,3,5,6,7] => [4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => ? = 3 + 2
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [9,7,1,2,6,3,8,4,5] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
 => [9,4,1,6,2,8,3,5,7] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
 => [2,6,9,1,3,4,5,7,8] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
 => [2,9,5,1,6,7,8,3,4] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
 => [9,3,7,1,2,4,5,6,8] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
 => [6,9,4,1,2,3,5,7,8] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
 => [6,8,4,1,2,3,5,9,7] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => [8,5,4,1,6,7,2,9,3] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => [9,5,4,1,6,7,8,2,3] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,0,0,0,1,0,1,0,1,0,0,1,0]
 => [2,3,7,9,1,4,5,6,8] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,0,0,0,1,1,1,0,1,0,0,0,0]
 => [2,3,9,6,1,7,8,4,5] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
 => [2,9,6,5,1,7,8,3,4] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
 => [9,3,4,8,1,2,5,6,7] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [7,8,9,6,1,2,3,4,5] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
 => [8,3,6,5,1,7,2,9,4] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
 => [8,3,4,7,6,1,2,9,5] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [9,3,7,5,6,1,8,2,4] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0]
 => [2,3,9,5,6,8,1,4,7] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => [2,9,4,5,6,8,1,3,7] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => [2,9,4,5,8,7,1,3,6] => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? = 4 + 2
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [10,1,9,5,6,7,8,2,3,4] => [8,2]
 => [[1,2,5,6,7,8,9,10],[3,4]]
 => ? = 2 + 2
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [11,1,10,5,6,7,8,9,2,3,4] => [9,2]
 => [[1,2,5,6,7,8,9,10,11],[3,4]]
 => ? = 2 + 2
[1,0,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [3,1,10,9,6,7,8,2,4,5] => [8,2]
 => [[1,2,5,6,7,8,9,10],[3,4]]
 => ? = 2 + 2
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [10,6,1,2,3,4,9,5,7,8] => [5,3,2]
 => [[1,2,5,9,10],[3,4,8],[6,7]]
 => ? = 3 + 2
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [10,7,1,2,3,4,9,5,6,8] => [5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => ? = 5 + 2
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [7,6,1,2,3,4,10,9,5,8] => [7,3]
 => [[1,2,3,7,8,9,10],[4,5,6]]
 => ? = 3 + 2
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [10,9,1,2,8,7,3,4,5,6] => [5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => ? = 5 + 2
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [10,8,1,9,2,3,4,5,6,7] => [7,3]
 => [[1,2,3,7,8,9,10],[4,5,6]]
 => ? = 3 + 2
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [9,8,1,10,7,2,3,4,5,6] => [5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => ? = 5 + 2
[1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [10,4,1,5,6,7,8,9,2,3] => [7,3]
 => [[1,2,3,7,8,9,10],[4,5,6]]
 => ? = 3 + 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [10,7,9,1,2,3,4,5,6,8] => [7,3]
 => [[1,2,3,7,8,9,10],[4,5,6]]
 => ? = 3 + 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [10,9,8,1,2,3,4,5,6,7] => [6,4]
 => [[1,2,3,4,9,10],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [7,8,9,10,1,2,3,4,5,6] => [5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => ? = 5 + 2
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [7,8,9,6,1,2,3,4,10,5] => [6,4]
 => [[1,2,3,4,9,10],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
 => [2,8,9,10,7,1,3,4,5,6] => [6,4]
 => [[1,2,3,4,9,10],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
 => [9,7,4,5,6,1,8,2,10,3] => [7,3]
 => [[1,2,3,7,8,9,10],[4,5,6]]
 => ? = 3 + 2
[1,1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0,0]
 => [10,3,4,7,6,1,8,9,2,5] => [6,4]
 => [[1,2,3,4,9,10],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0]
 => [2,10,8,5,6,7,1,9,3,4] => [7,3]
 => [[1,2,3,7,8,9,10],[4,5,6]]
 => ? = 3 + 2
[1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0]
 => [9,3,8,5,6,7,1,2,10,4] => [7,3]
 => [[1,2,3,7,8,9,10],[4,5,6]]
 => ? = 3 + 2
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [9,8,4,5,6,7,1,2,10,3] => [8,2]
 => [[1,2,5,6,7,8,9,10],[3,4]]
 => ? = 2 + 2
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => [10,3,8,5,6,7,1,9,2,4] => [6,4]
 => [[1,2,3,4,9,10],[5,6,7,8]]
 => ? = 4 + 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0]
 => [2,10,4,5,6,7,9,1,3,8] => [6,4]
 => [[1,2,3,4,9,10],[5,6,7,8]]
 => ? = 4 + 2
Description
The last entry on the main diagonal of a standard tableau.
Matching statistic: St000657
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 60%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => {{1,3,5},{2,4}}
 => [3,2] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => {{1,2,4,6},{3,5}}
 => [4,2] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => {{1,3,5,6},{2,4}}
 => [4,2] => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => {{1,3,5},{2,4,6}}
 => [3,3] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => {{1,3,5,6},{2,4}}
 => [4,2] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => {{1,3,6},{2,4,5}}
 => [3,3] => 3
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => {{1,2,4,6},{3,5}}
 => [4,2] => 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => {{1,4,6},{2,3,5}}
 => [3,3] => 3
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => {{1,3,4,6},{2,5}}
 => [4,2] => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => {{1,2,3,5,7},{4,6}}
 => [5,2] => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => {{1,2,4,6,7},{3,5}}
 => [5,2] => 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => {{1,2,4,6},{3,5,7}}
 => [4,3] => 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => {{1,2,4,6,7},{3,5}}
 => [5,2] => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => {{1,2,4,7},{3,5,6}}
 => [4,3] => 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => {{1,2,3,5,7},{4,6}}
 => [5,2] => 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => {{1,2,5,7},{3,4,6}}
 => [4,3] => 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => {{1,2,4,5,7},{3,6}}
 => [5,2] => 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => {{1,2,3,5,7},{4,6}}
 => [5,2] => 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => {{1,3,5,6,7},{2,4}}
 => [5,2] => 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => {{1,3,5,6,7},{2,4}}
 => [5,2] => 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => {{1,3,5},{2,4,6,7}}
 => [3,4] => 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => {{1,3,5,7},{2,4,6}}
 => [4,3] => 3
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => {{1,3,5},{2,4,6,7}}
 => [3,4] => 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => {{1,3,5,6,7},{2,4}}
 => [5,2] => 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => {{1,3,5,6,7},{2,4}}
 => [5,2] => 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => {{1,3,5,6},{2,4,7}}
 => [4,3] => 3
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => {{1,3,5,6,7},{2,4}}
 => [5,2] => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => {{1,2,3,5,7},{4,6}}
 => [5,2] => 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => {{1,3,6,7},{2,4,5}}
 => [4,3] => 3
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => {{1,3,6},{2,4,5,7}}
 => [3,4] => 3
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => {{1,3,6,7},{2,4,5}}
 => [4,3] => 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => {{1,3,7},{2,4,5,6}}
 => [3,4] => 3
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => {{1,2,4,6,7},{3,5}}
 => [5,2] => 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => {{1,2,4,6},{3,5,7}}
 => [4,3] => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => {{1,2,4,6,7},{3,5}}
 => [5,2] => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => {{1,2,4,7},{3,5,6}}
 => [4,3] => 3
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => {{1,4,6,7},{2,3,5}}
 => [4,3] => 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => {{1,4,6},{2,3,5,7}}
 => [3,4] => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => {{1,4,6,7},{2,3,5}}
 => [4,3] => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => {{1,3,4,6,7},{2,5}}
 => [5,2] => 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => {{1,3,4,6},{2,5,7}}
 => [4,3] => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => {{1,3,4,6,7},{2,5}}
 => [5,2] => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => {{1,4,7},{2,3,5,6}}
 => [3,4] => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => {{1,3,4,7},{2,5,6}}
 => [4,3] => 3
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => {{1,2,3,5,7},{4,6}}
 => [5,2] => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => {{1,2,5,7},{3,4,6}}
 => [4,3] => 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => {{1,2,4,5,7},{3,6}}
 => [5,2] => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => {{1,5,7},{2,3,4,6}}
 => [3,4] => 3
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => {{1,4,5,7},{2,3,6}}
 => [4,3] => 3
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => {{1,3,4,5,7},{2,6}}
 => [5,2] => 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [8,1,5,2,6,7,3,4] => {{1,2,4,8},{3,5,6,7}}
 => ? => ? = 4
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [8,4,1,5,6,2,3,7] => {{1,3,7,8},{2,4,5,6}}
 => ? => ? = 4
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [7,4,1,5,6,2,8,3] => {{1,3,7,8},{2,4,5,6}}
 => ? => ? = 4
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [2,8,5,1,6,7,3,4] => {{1,2,4,8},{3,5,6,7}}
 => ? => ? = 4
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [8,6,4,5,1,7,2,3] => {{1,3,4,5,8},{2,6,7}}
 => ? => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [9,1,2,3,4,8,5,6,7] => {{1,2,3,4,5,7,9},{6,8}}
 => ? => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [8,1,2,3,9,4,5,6,7] => {{1,2,3,4,6,8},{5,7,9}}
 => ? => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [5,1,2,3,9,8,4,6,7] => {{1,2,3,4,5,7,9},{6,8}}
 => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [9,1,2,3,6,8,4,5,7] => {{1,2,3,4,7,9},{5,6,8}}
 => ? => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,1,2,9,3,8,5,6,7] => {{1,2,3,4,5,7,9},{6,8}}
 => ? => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [9,1,2,8,3,4,5,6,7] => {{1,2,3,5,7,9},{4,6,8}}
 => ? => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [7,1,2,8,3,4,5,9,6] => {{1,2,3,5,7},{4,6,8,9}}
 => ? => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,1,2,8,9,3,5,6,7] => {{1,2,3,4,6,8},{5,7,9}}
 => ? => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [8,1,2,5,9,3,4,6,7] => {{1,2,3,6,8},{4,5,7,9}}
 => ? => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [8,1,2,9,6,3,4,5,7] => {{1,2,3,5,6,8},{4,7,9}}
 => ? => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,1,2,5,9,8,3,6,7] => {{1,2,3,4,5,7,9},{6,8}}
 => ? => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [9,1,2,8,6,7,3,4,5] => {{1,2,3,5,6,7,9},{4,8}}
 => ? => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [8,1,7,2,3,4,5,9,6] => {{1,2,4,6,8,9},{3,5,7}}
 => ? => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [3,1,9,8,2,4,5,6,7] => {{1,2,3,5,7,9},{4,6,8}}
 => ? => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [7,1,4,9,2,3,5,6,8] => {{1,2,5,7},{3,4,6,8,9}}
 => ? => ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [9,1,4,8,2,3,5,6,7] => {{1,2,5,7,9},{3,4,6,8}}
 => ? => ? = 4
[1,0,1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [9,1,6,5,2,3,4,7,8] => {{1,2,4,5,7,8,9},{3,6}}
 => ? => ? = 2
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [3,1,4,8,9,2,5,6,7] => {{1,2,3,4,6,8},{5,7,9}}
 => ? => ? = 3
[1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [8,1,9,5,6,2,3,4,7] => {{1,2,4,5,6,8},{3,7,9}}
 => ? => ? = 3
[1,0,1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [3,1,4,9,8,7,2,5,6] => {{1,2,3,4,6,7,9},{5,8}}
 => ? => ? = 2
[1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,1,9,8,6,7,2,4,5] => {{1,2,3,5,6,7,9},{4,8}}
 => ? => ? = 2
[1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [9,1,4,8,6,7,2,3,5] => {{1,2,5,6,7,9},{3,4,8}}
 => ? => ? = 3
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [9,1,8,5,6,7,2,3,4] => {{1,2,4,5,6,7,9},{3,8}}
 => ? => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,9,1,3,4,8,5,6,7] => {{1,2,3,4,5,7,9},{6,8}}
 => ? => ? = 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,8,1,3,9,4,5,6,7] => {{1,2,3,4,6,8},{5,7,9}}
 => ? => ? = 3
[1,1,0,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,9,1,3,8,7,4,5,6] => {{1,2,3,4,6,7,9},{5,8}}
 => ? => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,9,1,8,3,4,5,6,7] => {{1,2,3,5,7,9},{4,6,8}}
 => ? => ? = 3
[1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,7,1,6,3,4,8,9,5] => {{1,2,3,5,7,8,9},{4,6}}
 => ? => ? = 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,4,1,8,9,3,5,6,7] => {{1,2,3,4,6,8},{5,7,9}}
 => ? => ? = 3
[1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,9,1,8,6,7,3,4,5] => {{1,2,3,5,6,7,9},{4,8}}
 => ? => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [9,4,1,2,3,5,6,7,8] => {{1,3,5,6,7,8,9},{2,4}}
 => ? => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [7,4,1,2,3,5,8,9,6] => {{1,3,5,6,7,8,9},{2,4}}
 => ? => ? = 2
[1,1,0,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [6,4,1,2,3,7,8,9,5] => {{1,3,5,6,7,8,9},{2,4}}
 => ? => ? = 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [5,9,1,2,3,4,6,7,8] => {{1,3,5},{2,4,6,7,8,9}}
 => ? => ? = 3
[1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [5,8,1,2,3,4,6,9,7] => {{1,3,5},{2,4,6,7,8,9}}
 => ? => ? = 3
[1,1,0,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [5,7,1,2,3,4,8,9,6] => {{1,3,5},{2,4,6,7,8,9}}
 => ? => ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [9,6,1,2,3,4,5,7,8] => {{1,3,5,7,8,9},{2,4,6}}
 => ? => ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [8,6,1,2,3,4,5,9,7] => {{1,3,5,7,8,9},{2,4,6}}
 => ? => ? = 3
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [9,6,1,2,3,4,8,5,7] => {{1,3,5,7,8,9},{2,4,6}}
 => ? => ? = 3
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [7,6,1,2,3,4,8,9,5] => {{1,3,5,7,8,9},{2,4,6}}
 => ? => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [5,9,1,2,3,8,4,6,7] => {{1,3,5},{2,4,7,9},{6,8}}
 => ? => ? = 3
[1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [8,6,1,2,3,7,4,9,5] => {{1,3,5,8,9},{2,4,6,7}}
 => ? => ? = 4
[1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [9,6,1,2,3,7,8,4,5] => {{1,3,5,9},{2,4,6,7,8}}
 => ? => ? = 4
[1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [5,6,1,2,3,7,8,9,4] => {{1,3,5},{2,4,6,7,8,9}}
 => ? => ? = 3
[1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,1,2,9,3,6,7,8] => {{1,3,5,6,7,8,9},{2,4}}
 => ? => ? = 2
Description
The smallest part of an integer composition.
Matching statistic: St001180
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001180: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 60%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 5 = 3 + 2
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4 = 2 + 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 5 = 3 + 2
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 4 = 2 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [8,1,2,3,7,4,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [8,1,2,6,3,4,5,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [7,1,2,6,3,4,8,5] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,1,2,8,7,3,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [8,1,2,7,6,3,4,5] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [3,1,8,2,7,4,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [8,1,5,2,3,4,6,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,1,5,2,3,4,8,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [6,1,5,2,3,8,4,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [8,1,5,2,3,7,4,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [6,1,5,2,3,7,8,4] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [8,1,4,2,7,3,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [3,1,8,6,2,4,5,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,7,6,2,4,8,5] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [8,1,6,5,2,3,4,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [7,1,6,5,2,3,8,4] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [3,1,4,8,7,2,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,1,8,7,6,2,4,5] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [8,1,7,5,6,2,3,4] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [2,8,1,3,7,4,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,8,1,6,3,4,5,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [2,7,1,6,3,4,8,5] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [2,4,1,8,7,3,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [2,8,1,7,6,3,4,5] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [8,3,1,2,7,4,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [8,4,1,2,3,5,6,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [7,4,1,2,3,5,8,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,1,2,3,8,5,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [8,4,1,2,3,7,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [6,4,1,2,3,7,8,5] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,8,3,6,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => [5,4,1,2,7,3,8,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [8,4,1,2,6,3,5,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [7,4,1,2,6,3,8,5] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [5,4,1,2,6,8,3,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [5,4,1,2,8,7,3,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [8,4,1,2,6,7,3,5] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [5,4,1,2,6,7,8,3] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [8,3,1,6,2,4,5,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [7,3,1,6,2,4,8,5] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,1,8,7,2,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [8,3,1,7,6,2,4,5] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [2,3,8,1,7,4,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [2,8,5,1,3,4,6,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
 => [2,7,5,1,3,4,8,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => [2,6,5,1,3,8,4,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [2,8,5,1,3,7,4,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [2,6,5,1,3,7,8,4] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [2,8,4,1,7,3,5,6] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [8,5,4,1,2,3,6,7] => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => ? = 2 + 2
Description
Number of indecomposable injective modules with projective dimension at most 1.
Matching statistic: St001075
(load all 4 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 40%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => {{1,3,5},{2,4}}
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => {{1,2,4,6},{3,5}}
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => {{1,3,5,6},{2,4}}
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => {{1,3,5},{2,4,6}}
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => {{1,3,5,6},{2,4}}
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => {{1,3,6},{2,4,5}}
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => {{1,2,4,6},{3,5}}
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => {{1,4,6},{2,3,5}}
 => 3
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => {{1,3,4,6},{2,5}}
 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => {{1,2,3,5,7},{4,6}}
 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => {{1,2,4,6,7},{3,5}}
 => 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => {{1,2,4,6},{3,5,7}}
 => 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => {{1,2,4,6,7},{3,5}}
 => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => {{1,2,4,7},{3,5,6}}
 => 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => {{1,2,3,5,7},{4,6}}
 => 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => {{1,2,5,7},{3,4,6}}
 => 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => {{1,2,4,5,7},{3,6}}
 => 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => {{1,2,3,5,7},{4,6}}
 => 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => {{1,3,5,6,7},{2,4}}
 => 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => {{1,3,5,6,7},{2,4}}
 => 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => {{1,3,5},{2,4,6,7}}
 => 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => {{1,3,5,7},{2,4,6}}
 => 3
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => {{1,3,5},{2,4,6,7}}
 => 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => {{1,3,5,6,7},{2,4}}
 => 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => {{1,3,5,6,7},{2,4}}
 => 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => {{1,3,5,6},{2,4,7}}
 => 3
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => {{1,3,5,6,7},{2,4}}
 => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => {{1,2,3,5,7},{4,6}}
 => 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => {{1,3,6,7},{2,4,5}}
 => 3
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => {{1,3,6},{2,4,5,7}}
 => 3
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => {{1,3,6,7},{2,4,5}}
 => 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => {{1,3,7},{2,4,5,6}}
 => 3
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => {{1,2,4,6,7},{3,5}}
 => 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => {{1,2,4,6},{3,5,7}}
 => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => {{1,2,4,6,7},{3,5}}
 => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => {{1,2,4,7},{3,5,6}}
 => 3
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => {{1,4,6,7},{2,3,5}}
 => 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => {{1,4,6},{2,3,5,7}}
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => {{1,4,6,7},{2,3,5}}
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => {{1,3,4,6,7},{2,5}}
 => 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => {{1,3,4,6},{2,5,7}}
 => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => {{1,3,4,6,7},{2,5}}
 => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => {{1,4,7},{2,3,5,6}}
 => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => {{1,3,4,7},{2,5,6}}
 => 3
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => {{1,2,3,5,7},{4,6}}
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => {{1,2,5,7},{3,4,6}}
 => 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => {{1,2,4,5,7},{3,6}}
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => {{1,5,7},{2,3,4,6}}
 => 3
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => {{1,4,5,7},{2,3,6}}
 => 3
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => {{1,3,4,5,7},{2,6}}
 => 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [7,1,2,8,3,4,5,6] => {{1,2,3,5,7},{4,6,8}}
 => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [8,1,2,6,3,7,4,5] => {{1,2,3,5,8},{4,6,7}}
 => ? = 3
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [8,1,2,5,7,3,4,6] => {{1,2,3,6,8},{4,5,7}}
 => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [6,1,8,2,3,4,5,7] => {{1,2,4,6},{3,5,7,8}}
 => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [8,1,7,2,3,4,5,6] => {{1,2,4,6,8},{3,5,7}}
 => ? = 3
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [6,1,7,2,3,4,8,5] => {{1,2,4,6},{3,5,7,8}}
 => ? = 4
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [6,1,8,2,3,7,4,5] => {{1,2,4,6,7},{3,5,8}}
 => ? = 3
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [8,1,5,2,6,3,4,7] => {{1,2,4,7,8},{3,5,6}}
 => ? = 3
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [7,1,8,2,6,3,4,5] => {{1,2,4,7},{3,5,6,8}}
 => ? = 4
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [7,1,5,2,6,3,8,4] => {{1,2,4,7,8},{3,5,6}}
 => ? = 3
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [8,1,5,2,6,7,3,4] => {{1,2,4,8},{3,5,6,7}}
 => ? = 4
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,7,8,2,4,5,6] => {{1,2,3,5,7},{4,6,8}}
 => ? = 3
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,1,8,6,2,7,4,5] => {{1,2,3,5,8},{4,6,7}}
 => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [8,1,4,6,2,3,5,7] => {{1,2,5,7,8},{3,4,6}}
 => ? = 3
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,1,4,8,2,3,5,6] => {{1,2,5,7},{3,4,6,8}}
 => ? = 4
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [7,1,4,6,2,3,8,5] => {{1,2,5,7,8},{3,4,6}}
 => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [7,1,8,5,2,3,4,6] => {{1,2,4,5,7},{3,6,8}}
 => ? = 3
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [8,1,4,6,2,7,3,5] => {{1,2,5,8},{3,4,6,7}}
 => ? = 4
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [8,1,6,5,2,7,3,4] => {{1,2,4,5,8},{3,6,7}}
 => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [3,1,8,5,7,2,4,6] => {{1,2,3,6,8},{4,5,7}}
 => ? = 3
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [8,1,4,5,7,2,3,6] => {{1,2,6,8},{3,4,5,7}}
 => ? = 4
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [8,1,4,7,6,2,3,5] => {{1,2,5,6,8},{3,4,7}}
 => ? = 3
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [2,7,1,8,3,4,5,6] => {{1,2,3,5,7},{4,6,8}}
 => ? = 3
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [2,8,1,5,7,3,4,6] => {{1,2,3,6,8},{4,5,7}}
 => ? = 3
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [5,8,1,2,3,4,6,7] => {{1,3,5},{2,4,6,7,8}}
 => ? = 3
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [5,7,1,2,3,4,8,6] => {{1,3,5},{2,4,6,7,8}}
 => ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [8,6,1,2,3,4,5,7] => {{1,3,5,7,8},{2,4,6}}
 => ? = 3
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [7,6,1,2,3,4,8,5] => {{1,3,5,7,8},{2,4,6}}
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [5,6,1,2,3,8,4,7] => {{1,3,5},{2,4,6,7,8}}
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [5,8,1,2,3,7,4,6] => {{1,3,5},{2,4,6,7,8}}
 => ? = 3
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [8,6,1,2,3,7,4,5] => {{1,3,5,8},{2,4,6,7}}
 => ? = 4
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [5,6,1,2,3,7,8,4] => {{1,3,5},{2,4,6,7,8}}
 => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [8,4,1,2,7,3,5,6] => {{1,3,6,8},{2,4},{5,7}}
 => ? = 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [5,8,1,2,6,3,4,7] => {{1,3,5,6},{2,4,7,8}}
 => ? = 4
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [8,7,1,2,6,3,4,5] => {{1,3,5,6,8},{2,4,7}}
 => ? = 3
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [5,7,1,2,6,3,8,4] => {{1,3,5,6},{2,4,7,8}}
 => ? = 4
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [5,8,1,2,6,7,3,4] => {{1,3,5,6,7},{2,4,8}}
 => ? = 3
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [7,3,1,8,2,4,5,6] => {{1,2,3,5,7},{4,6,8}}
 => ? = 3
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [8,3,1,6,2,7,4,5] => {{1,2,3,5,8},{4,6,7}}
 => ? = 3
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [8,4,1,5,2,3,6,7] => {{1,3,6,7,8},{2,4,5}}
 => ? = 3
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => [7,4,1,5,2,3,8,6] => {{1,3,6,7,8},{2,4,5}}
 => ? = 3
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => [6,8,1,5,2,3,4,7] => {{1,3,6},{2,4,5,7,8}}
 => ? = 3
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [8,7,1,5,2,3,4,6] => {{1,3,6,8},{2,4,5,7}}
 => ? = 4
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [6,7,1,5,2,3,8,4] => {{1,3,6},{2,4,5,7,8}}
 => ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => [6,4,1,5,2,8,3,7] => {{1,3,6,7,8},{2,4,5}}
 => ? = 3
[1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => [8,4,1,5,2,7,3,6] => {{1,3,6,7,8},{2,4,5}}
 => ? = 3
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [6,8,1,5,2,7,3,4] => {{1,3,6,7},{2,4,5,8}}
 => ? = 4
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [6,4,1,5,2,7,8,3] => {{1,3,6,7,8},{2,4,5}}
 => ? = 3
[1,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => [8,3,1,5,7,2,4,6] => {{1,2,3,6,8},{4,5,7}}
 => ? = 3
[1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [8,4,1,5,6,2,3,7] => {{1,3,7,8},{2,4,5,6}}
 => ? = 4
Description
The minimal size of a block of a set partition.
Matching statistic: St000253
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00219: Set partitions —inverse Yip⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 40%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => {{1,3,5},{2,4}}
 => {{1,4,5},{2,3}}
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => {{1,2,4,6},{3,5}}
 => {{1,2,5,6},{3,4}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => {{1,3,5,6},{2,4}}
 => {{1,4,5,6},{2,3}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => {{1,3,5},{2,4,6}}
 => {{1,4,5},{2,3,6}}
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => {{1,3,5,6},{2,4}}
 => {{1,4,5,6},{2,3}}
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => {{1,3,6},{2,4,5}}
 => {{1,4,6},{2,3,5}}
 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => {{1,2,4,6},{3,5}}
 => {{1,2,5,6},{3,4}}
 => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => {{1,4,6},{2,3,5}}
 => {{1,3,5},{2,4,6}}
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => {{1,3,4,6},{2,5}}
 => {{1,5,6},{2,3,4}}
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => {{1,2,3,5,7},{4,6}}
 => {{1,2,3,6,7},{4,5}}
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => {{1,2,4,6,7},{3,5}}
 => {{1,2,5,6,7},{3,4}}
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => {{1,2,4,6},{3,5,7}}
 => {{1,2,5,6},{3,4,7}}
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => {{1,2,4,6,7},{3,5}}
 => {{1,2,5,6,7},{3,4}}
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => {{1,2,4,7},{3,5,6}}
 => {{1,2,5},{3,4,6,7}}
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => {{1,2,3,5,7},{4,6}}
 => {{1,2,3,6,7},{4,5}}
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => {{1,2,5,7},{3,4,6}}
 => {{1,2,4,5},{3,6,7}}
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => {{1,2,4,5,7},{3,6}}
 => {{1,2,6,7},{3,4,5}}
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => {{1,2,3,5,7},{4,6}}
 => {{1,2,3,6,7},{4,5}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => {{1,3,5,6,7},{2,4}}
 => {{1,4,5,6,7},{2,3}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => {{1,3,5,6,7},{2,4}}
 => {{1,4,5,6,7},{2,3}}
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => {{1,3,5},{2,4,6,7}}
 => {{1,4,5,7},{2,3,6}}
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => {{1,3,5,7},{2,4,6}}
 => {{1,4,5},{2,3,6,7}}
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => {{1,3,5},{2,4,6,7}}
 => {{1,4,5,7},{2,3,6}}
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => {{1,3,5,6,7},{2,4}}
 => {{1,4,5,6,7},{2,3}}
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => {{1,3,5,6,7},{2,4}}
 => {{1,4,5,6,7},{2,3}}
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => {{1,3,5,6},{2,4,7}}
 => {{1,4,5,6},{2,3,7}}
 => 2 = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => {{1,3,5,6,7},{2,4}}
 => {{1,4,5,6,7},{2,3}}
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => {{1,2,3,5,7},{4,6}}
 => {{1,2,3,6,7},{4,5}}
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => {{1,3,6,7},{2,4,5}}
 => {{1,4,6,7},{2,3,5}}
 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => {{1,3,6},{2,4,5,7}}
 => {{1,4,6},{2,3,5,7}}
 => 2 = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => {{1,3,6,7},{2,4,5}}
 => {{1,4,6,7},{2,3,5}}
 => 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => {{1,3,7},{2,4,5,6}}
 => {{1,4,7},{2,3,5,6}}
 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => {{1,2,4,6,7},{3,5}}
 => {{1,2,5,6,7},{3,4}}
 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => {{1,2,4,6},{3,5,7}}
 => {{1,2,5,6},{3,4,7}}
 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => {{1,2,4,6,7},{3,5}}
 => {{1,2,5,6,7},{3,4}}
 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => {{1,2,4,7},{3,5,6}}
 => {{1,2,5},{3,4,6,7}}
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => {{1,4,6,7},{2,3,5}}
 => {{1,3,5},{2,4,6,7}}
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => {{1,4,6},{2,3,5,7}}
 => {{1,3,5,7},{2,4,6}}
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => {{1,4,6,7},{2,3,5}}
 => {{1,3,5},{2,4,6,7}}
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => {{1,3,4,6,7},{2,5}}
 => {{1,5,6,7},{2,3,4}}
 => 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => {{1,3,4,6},{2,5,7}}
 => {{1,5,6},{2,3,4,7}}
 => 2 = 3 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => {{1,3,4,6,7},{2,5}}
 => {{1,5,6,7},{2,3,4}}
 => 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => {{1,4,7},{2,3,5,6}}
 => {{1,3,5,6},{2,4,7}}
 => 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => {{1,3,4,7},{2,5,6}}
 => {{1,5},{2,3,4,6,7}}
 => 2 = 3 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => {{1,2,3,5,7},{4,6}}
 => {{1,2,3,6,7},{4,5}}
 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => {{1,2,5,7},{3,4,6}}
 => {{1,2,4,5},{3,6,7}}
 => 2 = 3 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => {{1,2,4,5,7},{3,6}}
 => {{1,2,6,7},{3,4,5}}
 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => {{1,5,7},{2,3,4,6}}
 => {{1,3,4,7},{2,5,6}}
 => 2 = 3 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => {{1,4,5,7},{2,3,6}}
 => {{1,3,6,7},{2,4,5}}
 => 2 = 3 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => {{1,3,4,5,7},{2,6}}
 => {{1,6,7},{2,3,4,5}}
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [7,1,2,8,3,4,5,6] => {{1,2,3,5,7},{4,6,8}}
 => {{1,2,3,6,7},{4,5,8}}
 => ? = 3 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [8,1,2,6,3,7,4,5] => {{1,2,3,5,8},{4,6,7}}
 => {{1,2,3,6},{4,5,7,8}}
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [8,1,2,5,7,3,4,6] => {{1,2,3,6,8},{4,5,7}}
 => {{1,2,3,5,6},{4,7,8}}
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [8,1,2,7,6,3,4,5] => {{1,2,3,5,6,8},{4,7}}
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [6,1,8,2,3,4,5,7] => {{1,2,4,6},{3,5,7,8}}
 => {{1,2,5,6,8},{3,4,7}}
 => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [8,1,7,2,3,4,5,6] => {{1,2,4,6,8},{3,5,7}}
 => {{1,2,5,6},{3,4,7,8}}
 => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [6,1,7,2,3,4,8,5] => {{1,2,4,6},{3,5,7,8}}
 => {{1,2,5,6,8},{3,4,7}}
 => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [6,1,8,2,3,7,4,5] => {{1,2,4,6,7},{3,5,8}}
 => {{1,2,5,6,7},{3,4,8}}
 => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [8,1,5,2,6,3,4,7] => {{1,2,4,7,8},{3,5,6}}
 => {{1,2,5},{3,4,6,7,8}}
 => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [7,1,8,2,6,3,4,5] => {{1,2,4,7},{3,5,6,8}}
 => {{1,2,5,8},{3,4,6,7}}
 => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [7,1,5,2,6,3,8,4] => {{1,2,4,7,8},{3,5,6}}
 => {{1,2,5},{3,4,6,7,8}}
 => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [8,1,5,2,6,7,3,4] => {{1,2,4,8},{3,5,6,7}}
 => {{1,2,5,7},{3,4,6,8}}
 => ? = 4 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,7,8,2,4,5,6] => {{1,2,3,5,7},{4,6,8}}
 => {{1,2,3,6,7},{4,5,8}}
 => ? = 3 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,1,8,6,2,7,4,5] => {{1,2,3,5,8},{4,6,7}}
 => {{1,2,3,6},{4,5,7,8}}
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [8,1,4,6,2,3,5,7] => {{1,2,5,7,8},{3,4,6}}
 => {{1,2,4,5},{3,6,7,8}}
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,1,4,8,2,3,5,6] => {{1,2,5,7},{3,4,6,8}}
 => {{1,2,4,5,8},{3,6,7}}
 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [7,1,4,6,2,3,8,5] => {{1,2,5,7,8},{3,4,6}}
 => {{1,2,4,5},{3,6,7,8}}
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [8,1,6,5,2,3,4,7] => {{1,2,4,5,7,8},{3,6}}
 => {{1,2,6,7,8},{3,4,5}}
 => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [7,1,8,5,2,3,4,6] => {{1,2,4,5,7},{3,6,8}}
 => {{1,2,6,7},{3,4,5,8}}
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [7,1,6,5,2,3,8,4] => {{1,2,4,5,7,8},{3,6}}
 => {{1,2,6,7,8},{3,4,5}}
 => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [8,1,4,6,2,7,3,5] => {{1,2,5,8},{3,4,6,7}}
 => {{1,2,4,5,7},{3,6,8}}
 => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [8,1,6,5,2,7,3,4] => {{1,2,4,5,8},{3,6,7}}
 => {{1,2,6},{3,4,5,7,8}}
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [3,1,8,5,7,2,4,6] => {{1,2,3,6,8},{4,5,7}}
 => {{1,2,3,5,6},{4,7,8}}
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,1,8,7,6,2,4,5] => {{1,2,3,5,6,8},{4,7}}
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [8,1,4,5,7,2,3,6] => {{1,2,6,8},{3,4,5,7}}
 => {{1,2,4,6,8},{3,5,7}}
 => ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [8,1,4,7,6,2,3,5] => {{1,2,5,6,8},{3,4,7}}
 => {{1,2,4,5,6},{3,7,8}}
 => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [8,1,7,5,6,2,3,4] => {{1,2,4,5,6,8},{3,7}}
 => {{1,2,7,8},{3,4,5,6}}
 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [2,7,1,8,3,4,5,6] => {{1,2,3,5,7},{4,6,8}}
 => {{1,2,3,6,7},{4,5,8}}
 => ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [2,8,1,5,7,3,4,6] => {{1,2,3,6,8},{4,5,7}}
 => {{1,2,3,5,6},{4,7,8}}
 => ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [2,8,1,7,6,3,4,5] => {{1,2,3,5,6,8},{4,7}}
 => {{1,2,3,7,8},{4,5,6}}
 => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [5,8,1,2,3,4,6,7] => {{1,3,5},{2,4,6,7,8}}
 => {{1,4,5,7,8},{2,3,6}}
 => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [5,7,1,2,3,4,8,6] => {{1,3,5},{2,4,6,7,8}}
 => {{1,4,5,7,8},{2,3,6}}
 => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [8,6,1,2,3,4,5,7] => {{1,3,5,7,8},{2,4,6}}
 => {{1,4,5},{2,3,6,7,8}}
 => ? = 3 - 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [7,6,1,2,3,4,8,5] => {{1,3,5,7,8},{2,4,6}}
 => {{1,4,5},{2,3,6,7,8}}
 => ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [5,6,1,2,3,8,4,7] => {{1,3,5},{2,4,6,7,8}}
 => {{1,4,5,7,8},{2,3,6}}
 => ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [5,8,1,2,3,7,4,6] => {{1,3,5},{2,4,6,7,8}}
 => {{1,4,5,7,8},{2,3,6}}
 => ? = 3 - 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [8,6,1,2,3,7,4,5] => {{1,3,5,8},{2,4,6,7}}
 => {{1,4,5,7},{2,3,6,8}}
 => ? = 4 - 1
[1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [5,6,1,2,3,7,8,4] => {{1,3,5},{2,4,6,7,8}}
 => {{1,4,5,7,8},{2,3,6}}
 => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [8,4,1,2,7,3,5,6] => {{1,3,6,8},{2,4},{5,7}}
 => {{1,4},{2,3,7,8},{5,6}}
 => ? = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [5,8,1,2,6,3,4,7] => {{1,3,5,6},{2,4,7,8}}
 => {{1,4,5,6,8},{2,3,7}}
 => ? = 4 - 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [8,7,1,2,6,3,4,5] => {{1,3,5,6,8},{2,4,7}}
 => {{1,4,5,6},{2,3,7,8}}
 => ? = 3 - 1
[1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [5,7,1,2,6,3,8,4] => {{1,3,5,6},{2,4,7,8}}
 => {{1,4,5,6,8},{2,3,7}}
 => ? = 4 - 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [5,8,1,2,6,7,3,4] => {{1,3,5,6,7},{2,4,8}}
 => {{1,4,5,6,7},{2,3,8}}
 => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [7,3,1,8,2,4,5,6] => {{1,2,3,5,7},{4,6,8}}
 => {{1,2,3,6,7},{4,5,8}}
 => ? = 3 - 1
[1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => [8,3,1,6,2,7,4,5] => {{1,2,3,5,8},{4,6,7}}
 => {{1,2,3,6},{4,5,7,8}}
 => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [8,4,1,5,2,3,6,7] => {{1,3,6,7,8},{2,4,5}}
 => {{1,4,6,7,8},{2,3,5}}
 => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => [7,4,1,5,2,3,8,6] => {{1,3,6,7,8},{2,4,5}}
 => {{1,4,6,7,8},{2,3,5}}
 => ? = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => [6,8,1,5,2,3,4,7] => {{1,3,6},{2,4,5,7,8}}
 => {{1,4,6},{2,3,5,7,8}}
 => ? = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [8,7,1,5,2,3,4,6] => {{1,3,6,8},{2,4,5,7}}
 => {{1,4,6,8},{2,3,5,7}}
 => ? = 4 - 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [6,7,1,5,2,3,8,4] => {{1,3,6},{2,4,5,7,8}}
 => {{1,4,6},{2,3,5,7,8}}
 => ? = 3 - 1
Description
The crossing number of a set partition. This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St001596The number of two-by-two squares inside a skew partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001644The dimension of a graph. St001330The hat guessing number of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001712The number of natural descents of a standard Young tableau. St000454The largest eigenvalue of a graph if it is integral.