Processing math: 100%

Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000870
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: 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 product of the hook lengths of the diagonal cells in an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells (i,i) of a partition.
Matching statistic: St000744
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 50% ●values known / values provided: 65%●distinct values known / distinct values provided: 50%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [[1,2,3,4],[5,6]]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2]
 => [[1,2,3,4],[5,6]]
 => 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]]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [[1,2,3,4],[5,6]]
 => 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]]
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [[1,2,3,4],[5,6]]
 => 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]]
 => 3
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [[1,2,3,4],[5,6]]
 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [4,3]
 => [[1,2,3,4],[5,6,7]]
 => 3
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 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] => [5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 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] => [5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 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] => [5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 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] => [5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 4
[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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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] => [6,3]
 => [[1,2,3,4,5,6],[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] => [6,3]
 => [[1,2,3,4,5,6],[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] => [6,3]
 => [[1,2,3,4,5,6],[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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 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] => [4,3,2]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => ? = 4
[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]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 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] => [5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 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] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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]
 => [[1,2,3,4],[5,6,7],[8,9]]
 => ? = 4
[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]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 4
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [9,8,1,2,6,7,3,4,5] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [9,3,1,8,2,4,5,6,7] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [6,9,1,5,2,3,4,7,8] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 4
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [8,9,1,5,6,7,2,3,4] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [9,4,1,5,6,7,8,2,3] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 4
[1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
 => [2,9,7,1,3,4,5,6,8] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
 => [2,8,4,1,9,3,5,6,7] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 4
[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]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 4
[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]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 4
[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]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 4
[1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
 => [9,7,4,1,2,3,5,6,8] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
 => [9,8,5,1,7,2,3,4,6] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 4
[1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
 => [9,3,5,1,6,7,8,2,4] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
[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]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 4
[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]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? = 4
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => [2,3,9,8,1,4,5,6,7] => [6,3]
 => [[1,2,3,4,5,6],[7,8,9]]
 => ? = 3
Description
The length of the path to the largest entry in a standard Young tableau.
Matching statistic: St001480
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 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]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
Matching statistic: St001508
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001508: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 67%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [3,3]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [8,1,2,3,7,4,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [8,1,2,6,3,4,5,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [7,1,2,6,3,4,8,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,1,2,8,7,3,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [8,1,2,7,6,3,4,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [3,1,8,2,7,4,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [8,1,5,2,3,4,6,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,1,5,2,3,4,8,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [6,1,5,2,3,8,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [8,1,5,2,3,7,4,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [6,1,5,2,3,7,8,4] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [8,1,4,2,7,3,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [3,1,8,6,2,4,5,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,7,6,2,4,8,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [8,1,6,5,2,3,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [7,1,6,5,2,3,8,4] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [3,1,4,8,7,2,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,1,8,7,6,2,4,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [8,1,7,5,6,2,3,4] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [2,8,1,3,7,4,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,8,1,6,3,4,5,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [2,7,1,6,3,4,8,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [2,4,1,8,7,3,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [2,8,1,7,6,3,4,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [8,3,1,2,7,4,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [8,4,1,2,3,5,6,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [7,4,1,2,3,5,8,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,1,2,3,8,5,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [8,4,1,2,3,7,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [6,4,1,2,3,7,8,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,8,3,6,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => [5,4,1,2,7,3,8,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [8,4,1,2,6,3,5,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [7,4,1,2,6,3,8,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [5,4,1,2,6,8,3,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [5,4,1,2,8,7,3,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [8,4,1,2,6,7,3,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [5,4,1,2,6,7,8,3] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [8,3,1,6,2,4,5,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [7,3,1,6,2,4,8,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,1,1,0,0,0,1,0,1,0,0]
 => [4,3,1,8,7,2,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => [8,3,1,7,6,2,4,5] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [2,3,8,1,7,4,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [2,8,5,1,3,4,6,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
 => [2,7,5,1,3,4,8,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => [2,6,5,1,3,8,4,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
 => [2,8,5,1,3,7,4,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [2,6,5,1,3,7,8,4] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [2,8,4,1,7,3,5,6] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [8,5,4,1,2,3,6,7] => [6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2 - 1
Description
The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. Given two lattice paths U,L from (0,0) to (d,n−d), [1] describes a bijection between lattice paths weakly between U and L and subsets of {1,…,n} such that the set of all such subsets gives the standard complex of the lattice path matroid M[U,L]. This statistic gives the cardinality of the image of this bijection when a Dyck path is considered as a path weakly above the diagonal and relative to the diagonal boundary.
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: 33%
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}}
 => ? = 3
[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: 33%
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}}
 => ? = 3 - 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.
Matching statistic: St000614
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
St000614: Set partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => {{1,3,5},{2,4}}
 => {{1,4},{2,3,5}}
 => 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},{3,4,6}}
 => 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},{2,3,5,6}}
 => 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},{2,3,5,6}}
 => 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},{3,4,6}}
 => 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}}
 => {{1,3,4,6},{2,5}}
 => 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},{2,3,4,6}}
 => 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},{4,5,7}}
 => 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},{3,4,6,7}}
 => 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},{3,4,6,7}}
 => 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}}
 => {{1,2,5,7},{3,4,6}}
 => 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},{4,5,7}}
 => 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}}
 => {{1,2,4,5,7},{3,6}}
 => 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},{3,4,5,7}}
 => 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},{4,5,7}}
 => 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},{2,3,5,6,7}}
 => 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},{2,3,5,6,7}}
 => 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},{2,3,6,7}}
 => 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}}
 => {{1,4,5,7},{2,3,6}}
 => 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},{2,3,6,7}}
 => 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},{2,3,5,6,7}}
 => 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},{2,3,5,6,7}}
 => 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},{2,3,5,6,7}}
 => 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},{4,5,7}}
 => 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}}
 => {{1,4,7},{2,3,5,6}}
 => 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}}
 => {{1,4,6},{2,3,5,7}}
 => 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},{3,4,6,7}}
 => 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},{3,4,6,7}}
 => 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}}
 => {{1,2,5,7},{3,4,6}}
 => 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}}
 => {{1,3,4,6,7},{2,5}}
 => 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}}
 => {{1,3,4,7},{2,5,6}}
 => 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}}
 => {{1,3,4,6,7},{2,5}}
 => 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},{2,3,4,6,7}}
 => 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},{2,3,4,6,7}}
 => 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}}
 => {{1,3,4,6},{2,5,7}}
 => 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}}
 => {{1,5,7},{2,3,4,6}}
 => 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},{4,5,7}}
 => 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}}
 => {{1,2,4,5,7},{3,6}}
 => 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},{3,4,5,7}}
 => 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}}
 => {{1,3,6},{2,4,5,7}}
 => 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}}
 => {{1,3,4,5,7},{2,6}}
 => 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},{2,3,4,5,7}}
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [8,1,2,3,7,4,5,6] => {{1,2,3,4,6,8},{5,7}}
 => {{1,2,3,4,7},{5,6,8}}
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [8,1,2,6,3,4,5,7] => {{1,2,3,5,7,8},{4,6}}
 => {{1,2,3,6},{4,5,7,8}}
 => ? = 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
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [7,1,2,6,3,4,8,5] => {{1,2,3,5,7,8},{4,6}}
 => {{1,2,3,6},{4,5,7,8}}
 => ? = 2 - 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}}
 => {{1,2,3,6,8},{4,5,7}}
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,1,2,8,7,3,5,6] => {{1,2,3,4,6,8},{5,7}}
 => {{1,2,3,4,7},{5,6,8}}
 => ? = 2 - 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},{4,5,6,8}}
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [3,1,8,2,7,4,5,6] => {{1,2,3,4,6,8},{5,7}}
 => {{1,2,3,4,7},{5,6,8}}
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [8,1,5,2,3,4,6,7] => {{1,2,4,6,7,8},{3,5}}
 => {{1,2,5},{3,4,6,7,8}}
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,1,5,2,3,4,8,6] => {{1,2,4,6,7,8},{3,5}}
 => {{1,2,5},{3,4,6,7,8}}
 => ? = 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},{3,4,7,8}}
 => ? = 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}}
 => {{1,2,5,6,8},{3,4,7}}
 => ? = 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},{3,4,7,8}}
 => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [6,1,5,2,3,8,4,7] => {{1,2,4,6,7,8},{3,5}}
 => {{1,2,5},{3,4,6,7,8}}
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [8,1,5,2,3,7,4,6] => {{1,2,4,6,7,8},{3,5}}
 => {{1,2,5},{3,4,6,7,8}}
 => ? = 2 - 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
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [6,1,5,2,3,7,8,4] => {{1,2,4,6,7,8},{3,5}}
 => {{1,2,5},{3,4,6,7,8}}
 => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [8,1,4,2,7,3,5,6] => {{1,2,3,4,6,8},{5,7}}
 => {{1,2,3,4,7},{5,6,8}}
 => ? = 2 - 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}}
 => {{1,2,5,7,8},{3,4,6}}
 => ? = 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}}
 => {{1,2,5,7,8},{3,4,6}}
 => ? = 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
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [3,1,8,6,2,4,5,7] => {{1,2,3,5,7,8},{4,6}}
 => {{1,2,3,6},{4,5,7,8}}
 => ? = 2 - 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
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,7,6,2,4,8,5] => {{1,2,3,5,7,8},{4,6}}
 => {{1,2,3,6},{4,5,7,8}}
 => ? = 2 - 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}}
 => {{1,2,3,6,8},{4,5,7}}
 => ? = 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
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [8,1,6,5,2,3,4,7] => {{1,2,4,5,7,8},{3,6}}
 => {{1,2,6},{3,4,5,7,8}}
 => ? = 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},{3,4,5,7,8}}
 => ? = 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}}
 => {{1,2,6,8},{3,4,5,7}}
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [3,1,4,8,7,2,5,6] => {{1,2,3,4,6,8},{5,7}}
 => {{1,2,3,4,7},{5,6,8}}
 => ? = 2 - 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},{4,5,6,8}}
 => ? = 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}}
 => {{1,2,4,7},{3,5,6,8}}
 => ? = 4 - 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},{3,4,5,6,8}}
 => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [2,8,1,3,7,4,5,6] => {{1,2,3,4,6,8},{5,7}}
 => {{1,2,3,4,7},{5,6,8}}
 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,8,1,6,3,4,5,7] => {{1,2,3,5,7,8},{4,6}}
 => {{1,2,3,6},{4,5,7,8}}
 => ? = 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
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [2,7,1,6,3,4,8,5] => {{1,2,3,5,7,8},{4,6}}
 => {{1,2,3,6},{4,5,7,8}}
 => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [2,4,1,8,7,3,5,6] => {{1,2,3,4,6,8},{5,7}}
 => {{1,2,3,4,7},{5,6,8}}
 => ? = 2 - 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},{4,5,6,8}}
 => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [8,3,1,2,7,4,5,6] => {{1,2,3,4,6,8},{5,7}}
 => {{1,2,3,4,7},{5,6,8}}
 => ? = 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},{2,3,6,7,8}}
 => ? = 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},{2,3,6,7,8}}
 => ? = 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}}
 => {{1,4,5,7,8},{2,3,6}}
 => ? = 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}}
 => {{1,4,5,7,8},{2,3,6}}
 => ? = 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},{2,3,6,7,8}}
 => ? = 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},{2,3,6,7,8}}
 => ? = 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
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block.
Matching statistic: St001596
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [[3,2],[]]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [[4,2],[]]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,2]
 => [[4,2],[]]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => [[3,3],[]]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [[4,2],[]]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => [[3,3],[]]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [[4,2],[]]
 => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [3,3]
 => [[3,3],[]]
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [[4,2],[]]
 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [3,1,7,6,2,4,5] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,1,4,6,2,3,5] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [7,1,6,5,2,3,4] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,7,1,6,3,4,5] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [7,4,1,2,3,5,6] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,4,1,2,3,7,5] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,7,1,2,3,4,6] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [5,4,1,2,7,3,6] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [7,4,1,5,2,3,6] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [2,7,5,1,3,4,6] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [7,3,5,1,2,4,6] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [7,5,4,1,2,3,6] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [4,3]
 => [[4,3],[]]
 => 2 = 3 - 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [5,2]
 => [[5,2],[]]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [8,1,2,3,7,4,5,6] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [8,1,2,6,3,4,5,7] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [7,1,2,8,3,4,5,6] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [7,1,2,6,3,4,8,5] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [8,1,2,6,3,7,4,5] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,1,2,8,7,3,5,6] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [8,1,2,5,7,3,4,6] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [8,1,2,7,6,3,4,5] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [3,1,8,2,7,4,5,6] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [8,1,5,2,3,4,6,7] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,1,5,2,3,4,8,6] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [6,1,8,2,3,4,5,7] => [4,4]
 => [[4,4],[]]
 => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [8,1,7,2,3,4,5,6] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [6,1,7,2,3,4,8,5] => [4,4]
 => [[4,4],[]]
 => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [6,1,5,2,3,8,4,7] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [8,1,5,2,3,7,4,6] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [6,1,8,2,3,7,4,5] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [6,1,5,2,3,7,8,4] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [8,1,4,2,7,3,5,6] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [8,1,5,2,6,3,4,7] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [7,1,8,2,6,3,4,5] => [4,4]
 => [[4,4],[]]
 => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [7,1,5,2,6,3,8,4] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [8,1,5,2,6,7,3,4] => [4,4]
 => [[4,4],[]]
 => ? = 4 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [3,1,8,6,2,4,5,7] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,7,8,2,4,5,6] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,7,6,2,4,8,5] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,1,8,6,2,7,4,5] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [8,1,4,6,2,3,5,7] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,1,4,8,2,3,5,6] => [4,4]
 => [[4,4],[]]
 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [7,1,4,6,2,3,8,5] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [8,1,6,5,2,3,4,7] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [7,1,8,5,2,3,4,6] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [7,1,6,5,2,3,8,4] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [8,1,4,6,2,7,3,5] => [4,4]
 => [[4,4],[]]
 => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [8,1,6,5,2,7,3,4] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [3,1,4,8,7,2,5,6] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [3,1,8,5,7,2,4,6] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,1,8,7,6,2,4,5] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [8,1,4,5,7,2,3,6] => [4,4]
 => [[4,4],[]]
 => ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [8,1,4,7,6,2,3,5] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [8,1,7,5,6,2,3,4] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [2,8,1,3,7,4,5,6] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,8,1,6,3,4,5,7] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [2,7,1,8,3,4,5,6] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [2,7,1,6,3,4,8,5] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [2,4,1,8,7,3,5,6] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [2,8,1,5,7,3,4,6] => [5,3]
 => [[5,3],[]]
 => ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [2,8,1,7,6,3,4,5] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [8,3,1,2,7,4,5,6] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [8,4,1,2,3,5,6,7] => [6,2]
 => [[6,2],[]]
 => ? = 2 - 1
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: St001816
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [3,2]
 => [[1,2,5],[3,4]]
 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [4,2]
 => [[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] => [4,2]
 => [[1,2,5,6],[3,4]]
 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [3,3]
 => [[1,2,3],[4,5,6]]
 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2]
 => [[1,2,5,6],[3,4]]
 => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [3,3]
 => [[1,2,3],[4,5,6]]
 => 2 = 3 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [4,2]
 => [[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] => [3,3]
 => [[1,2,3],[4,5,6]]
 => 2 = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [4,2]
 => [[1,2,5,6],[3,4]]
 => 1 = 2 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 2 = 3 - 1
[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]]
 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [8,1,2,3,7,4,5,6] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [8,1,2,6,3,4,5,7] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [7,1,2,8,3,4,5,6] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [7,1,2,6,3,4,8,5] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [8,1,2,6,3,7,4,5] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [4,1,2,8,7,3,5,6] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [8,1,2,5,7,3,4,6] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [8,1,2,7,6,3,4,5] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [3,1,8,2,7,4,5,6] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [8,1,5,2,3,4,6,7] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,1,5,2,3,4,8,6] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [6,1,8,2,3,4,5,7] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [8,1,7,2,3,4,5,6] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [6,1,7,2,3,4,8,5] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => ? = 4 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [6,1,5,2,3,8,4,7] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [8,1,5,2,3,7,4,6] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [6,1,8,2,3,7,4,5] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [6,1,5,2,3,7,8,4] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [8,1,4,2,7,3,5,6] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [8,1,5,2,6,3,4,7] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [7,1,8,2,6,3,4,5] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => ? = 4 - 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [7,1,5,2,6,3,8,4] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [8,1,5,2,6,7,3,4] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => ? = 4 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [3,1,8,6,2,4,5,7] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,7,8,2,4,5,6] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [3,1,7,6,2,4,8,5] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,1,8,6,2,7,4,5] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [8,1,4,6,2,3,5,7] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,1,4,8,2,3,5,6] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [7,1,4,6,2,3,8,5] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [8,1,6,5,2,3,4,7] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [7,1,8,5,2,3,4,6] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [7,1,6,5,2,3,8,4] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [8,1,4,6,2,7,3,5] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => ? = 4 - 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [8,1,6,5,2,7,3,4] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [3,1,4,8,7,2,5,6] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [3,1,8,5,7,2,4,6] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,1,8,7,6,2,4,5] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [8,1,4,5,7,2,3,6] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => ? = 4 - 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [8,1,4,7,6,2,3,5] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [8,1,7,5,6,2,3,4] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [2,8,1,3,7,4,5,6] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [2,8,1,6,3,4,5,7] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [2,7,1,8,3,4,5,6] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [2,7,1,6,3,4,8,5] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,1,0,0,1,1,1,0,0,1,0,1,0,0]
 => [2,4,1,8,7,3,5,6] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [2,8,1,5,7,3,4,6] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => ? = 3 - 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [2,8,1,7,6,3,4,5] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [8,3,1,2,7,4,5,6] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [8,4,1,2,3,5,6,7] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => ? = 2 - 1
Description
Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition λ has dimension equal to the number of standard tableaux of shape λ. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape λ; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
Matching statistic: St001258
Mapping
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001258: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%
Values
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 5 = 3 + 2
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 5 = 3 + 2
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => ? = 3 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 3 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [5,4,2]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => ? = 3 + 2
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 4 = 2 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 3 + 2
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 3 + 2
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 3 + 2
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => ? = 3 + 2
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 2 + 2
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 3 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => ? = 3 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 3 + 2
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 3 + 2
[1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 + 2
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 3 + 2
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 2 + 2
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => ? = 3 + 2
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 3 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => ? = 3 + 2
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 3 + 2
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 3 + 2
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => ? = 3 + 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 5 = 3 + 2
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [5,4,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => ? = 2 + 2
[1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => ? = 3 + 2
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => ? = 3 + 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 5 = 3 + 2
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [6,5,4]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 3 + 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 2 + 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 3 + 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [6,5,3]
 => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [6,4,3]
 => [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 3 + 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2]
 => [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2]
 => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2]
 => [1,0,1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? = 4 + 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2]
 => [1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? = 3 + 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2]
 => [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
 => ? = 4 + 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2]
 => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 2 + 2
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 4 = 2 + 2
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => 5 = 3 + 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 4 = 2 + 2
[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]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 5 = 3 + 2
[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]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 4 = 2 + 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 4 = 2 + 2
[1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 4 = 2 + 2
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 4 = 2 + 2
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0,0,0]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 4 = 2 + 2
Description
Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. For at most 6 simple modules this statistic coincides with the injective dimension of the regular module as a bimodule.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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.