Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001232
Mapping
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0,1,0]
 => 1
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,1]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[3,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[2,2,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,1,1,1,1]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[4,1,1]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[3,3]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[2,2,2]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[2,2,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[1,1,1,1,1,1]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1
[2,2,2,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[2,2,1,1,1]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 7
[5,3]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[5,1,1,1]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2
[4,2,2]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[3,3,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[2,2,2,2]
 => [4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => 6
[2,2,2,1,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 7
[2,2,1,1,1,1]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 9
[6,3]
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 5
[4,4,1]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3
[3,3,1,1,1]
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 5
[2,2,2,2,1]
 => [5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[2,2,2,1,1,1]
 => [6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => 10
[7,3]
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => 7
[6,1,1,1,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => 2
[5,2,2,1]
 => [4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 6
[3,3,3,1]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 5
[3,3,1,1,1,1]
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => 7
[2,2,2,2,2]
 => [5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => 7
[2,2,2,2,1,1]
 => [6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => 9
[6,4,1]
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
[5,2,2,2]
 => [4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5
[4,4,1,1,1]
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3
[3,3,3,1,1]
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 4
[2,2,2,2,2,1]
 => [6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => 6
[7,4,1]
 => [6,3,3]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => 5
[6,2,2,1,1]
 => [5,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => 8
[5,5,1,1]
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 3
[5,3,3,1]
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4
[4,4,4]
 => [3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => 4
[4,4,1,1,1,1]
 => [6,3,3]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => 5
[3,3,3,3]
 => [4,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => 7
[3,3,3,1,1,1]
 => [6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => 7
[6,2,2,2,1]
 => [5,5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => 8
[4,4,2,2,1]
 => [5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5
[3,3,3,3,1]
 => [5,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => 6
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001000
Mapping
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1]
 => [1,0,1,0]
 => 1
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => 2
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => 1
[1,1,1]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 1
[3,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,1]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[2,2,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,1,1,1,1]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 1
[4,1,1]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[3,3]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[2,2,2]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 5
[2,2,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5
[1,1,1,1,1,1]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[2,2,2,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4
[2,2,1,1,1]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ? = 7
[5,3]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[5,1,1,1]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 2
[4,2,2]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[3,3,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3
[2,2,2,2]
 => [4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 6
[2,2,2,1,1]
 => [5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => ? = 7
[2,2,1,1,1,1]
 => [6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => ? = 9
[6,3]
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 5
[4,4,1]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 3
[3,3,1,1,1]
 => [5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 5
[2,2,2,2,1]
 => [5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 5
[2,2,2,1,1,1]
 => [6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => ? = 10
[7,3]
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => ? = 7
[6,1,1,1,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 2
[5,2,2,1]
 => [4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => ? = 6
[3,3,3,1]
 => [4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => ? = 5
[3,3,1,1,1,1]
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => ? = 7
[2,2,2,2,2]
 => [5,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => ? = 7
[2,2,2,2,1,1]
 => [6,1,1,1,1]
 => [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 9
[6,4,1]
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 3
[5,2,2,2]
 => [4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 5
[4,4,1,1,1]
 => [5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 3
[3,3,3,1,1]
 => [5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
[2,2,2,2,2,1]
 => [6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 6
[7,4,1]
 => [6,3,3]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => ? = 5
[6,2,2,1,1]
 => [5,5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => ? = 8
[5,5,1,1]
 => [4,4,4]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 3
[5,3,3,1]
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
[4,4,4]
 => [3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 4
[4,4,1,1,1,1]
 => [6,3,3]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => ? = 5
[3,3,3,3]
 => [4,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => ? = 7
[3,3,3,1,1,1]
 => [6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => ? = 7
[6,2,2,2,1]
 => [5,5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 8
[4,4,2,2,1]
 => [5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 5
[3,3,3,3,1]
 => [5,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => ? = 6
[7,5,1,1]
 => [6,4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
[6,4,4]
 => [5,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => ? = 5
[6,3,3,1,1]
 => [5,5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => ? = 6
[6,2,2,2,2]
 => [5,5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 6
[5,5,2,2]
 => [4,4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 7
[5,5,1,1,1,1]
 => [6,4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? = 3
[5,3,3,3]
 => [4,4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => ? = 7
[4,4,4,1,1]
 => [5,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => ? = 5
[3,3,3,3,1,1]
 => [6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 5
[7,4,4]
 => [6,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
[4,4,4,1,1,1]
 => [6,3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 4
[4,4,2,2,2,1]
 => [6,3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => ? = 8
[7,5,2,2]
 => [6,4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 7
[6,6,4]
 => [5,5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
[6,4,4,1,1]
 => [5,5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 4
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001200
Mapping
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 20%
Values
[1]
 => [1]
 => [1,0]
 => [1,0]
 => ? = 1
[2]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[1,1]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? = 1
[1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? = 1
[3,1]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 1
[2,2,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[4,1,1]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[2,2,2]
 => [3,1,1,1]
 => [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
[2,2,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 5
[1,1,1,1,1,1]
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[2,2,2,1]
 => [4,1,1,1]
 => [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]
 => ? = 4
[2,2,1,1,1]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 7
[5,3]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 3
[5,1,1,1]
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[4,2,2]
 => [3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => ? = 4
[3,3,1,1]
 => [4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => ? = 3
[2,2,2,2]
 => [4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 6
[2,2,2,1,1]
 => [5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 7
[2,2,1,1,1,1]
 => [6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 9
[6,3]
 => [5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 5
[4,4,1]
 => [3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[3,3,1,1,1]
 => [5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 5
[2,2,2,2,1]
 => [5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 5
[2,2,2,1,1,1]
 => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 10
[7,3]
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 7
[6,1,1,1,1]
 => [5,5]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 2
[5,2,2,1]
 => [4,4,1,1]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 6
[3,3,3,1]
 => [4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 5
[3,3,1,1,1,1]
 => [6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
 => ? = 7
[2,2,2,2,2]
 => [5,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[2,2,2,2,1,1]
 => [6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 9
[6,4,1]
 => [5,3,3]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? = 3
[5,2,2,2]
 => [4,4,1,1,1]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 5
[4,4,1,1,1]
 => [5,3,3]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => ? = 3
[3,3,3,1,1]
 => [5,2,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 4
[2,2,2,2,2,1]
 => [6,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[7,4,1]
 => [6,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
 => ? = 5
[6,2,2,1,1]
 => [5,5,1,1]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
 => ? = 8
[5,5,1,1]
 => [4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? = 3
[5,3,3,1]
 => [4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => ? = 4
[4,4,4]
 => [3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 4
[4,4,1,1,1,1]
 => [6,3,3]
 => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,1,0,0,0]
 => ? = 5
[3,3,3,3]
 => [4,2,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 7
[3,3,3,1,1,1]
 => [6,2,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 7
[6,2,2,2,1]
 => [5,5,1,1,1]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 8
[4,4,2,2,1]
 => [5,3,3,1,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0,1,0]
 => ? = 5
[3,3,3,3,1]
 => [5,2,2,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => ?
 => ? = 6
[7,5,1,1]
 => [6,4,4]
 => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => ? = 3
[6,4,4]
 => [5,3,3,3]
 => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,1,0,1,0,0,0]
 => ? = 5
[6,3,3,1,1]
 => [5,5,2,2]
 => [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
 => ? = 6
[6,2,2,2,2]
 => [5,5,1,1,1,1]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 6
[5,5,2,2]
 => [4,4,4,1,1]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => ? = 7
[5,5,1,1,1,1]
 => [6,4,4]
 => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => ? = 3
[5,3,3,3]
 => [4,4,2,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
 => ? = 7
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001948
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 30%
Values
[1]
 => [[1]]
 => [1] => [1] => ? = 1 - 1
[2]
 => [[1,2]]
 => [1,2] => [1,2] => 1 = 2 - 1
[1,1]
 => [[1],[2]]
 => [2,1] => [2,1] => 0 = 1 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 0 = 1 - 1
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,4,5,2,3] => 2 = 3 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [3,2,1,4,5,6] => ? = 2 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [1,2,6,3,4,5] => ? = 3 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [1,6,2,5,3,4] => ? = 5 - 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [1,5,4,6,2,3] => ? = 5 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 1 - 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [1,3,7,5,6,2,4] => ? = 4 - 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [1,6,5,4,7,2,3] => ? = 7 - 1
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => [1,2,6,3,4,5,7,8] => ? = 3 - 1
[5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => ? = 2 - 1
[4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8] => [1,6,2,5,3,4,7,8] => ? = 4 - 1
[3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [6,3,2,7,8,1,4,5] => [1,5,4,2,8,3,6,7] => ? = 3 - 1
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => [1,8,2,7,3,6,4,5] => ? = 6 - 1
[2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [7,5,3,8,2,6,1,4] => [1,3,7,8,5,6,2,4] => ? = 7 - 1
[2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => [7,5,4,3,2,8,1,6] => [1,7,6,5,4,8,2,3] => ? = 9 - 1
[6,3]
 => [[1,2,3,7,8,9],[4,5,6]]
 => [4,5,6,1,2,3,7,8,9] => [1,2,6,3,4,5,7,8,9] => ? = 5 - 1
[4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => [6,2,7,8,9,1,3,4,5] => [1,4,2,3,9,5,6,7,8] => ? = 3 - 1
[3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3],[4],[7]]
 => [7,4,3,2,8,9,1,5,6] => [1,6,5,4,2,9,3,7,8] => ? = 5 - 1
[2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => [8,6,9,4,7,2,5,1,3] => [1,3,9,4,8,6,7,2,5] => ? = 5 - 1
[2,2,2,1,1,1]
 => [[1,5],[2,7],[3,9],[4],[6],[8]]
 => [8,6,4,3,9,2,7,1,5] => [1,3,8,7,9,5,6,2,4] => ? = 10 - 1
[7,3]
 => [[1,2,3,7,8,9,10],[4,5,6]]
 => [4,5,6,1,2,3,7,8,9,10] => [1,2,6,3,4,5,7,8,9,10] => ? = 7 - 1
[6,1,1,1,1]
 => [[1,6,7,8,9,10],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7,8,9,10] => [5,4,3,2,1,6,7,8,9,10] => ? = 2 - 1
[5,2,2,1]
 => [[1,3,8,9,10],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3,8,9,10] => [1,3,7,5,6,2,4,8,9,10] => ? = 6 - 1
[3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8]]
 => [8,5,9,10,2,6,7,1,3,4] => [1,3,2,10,6,4,9,5,7,8] => ? = 5 - 1
[3,3,1,1,1,1]
 => [[1,6,7],[2,9,10],[3],[4],[5],[8]]
 => [8,5,4,3,2,9,10,1,6,7] => [1,7,6,5,4,2,10,3,8,9] => ? = 7 - 1
[2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10]]
 => [9,10,7,8,5,6,3,4,1,2] => [1,10,2,9,3,8,4,7,5,6] => ? = 7 - 1
[2,2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5,10],[7],[9]]
 => [9,7,5,10,3,8,2,6,1,4] => [1,3,5,10,8,9,6,7,2,4] => ? = 9 - 1
[6,4,1]
 => [[1,3,4,5,10,11],[2,7,8,9],[6]]
 => ? => ? => ? = 3 - 1
[5,2,2,2]
 => [[1,2,9,10,11],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2,9,10,11] => ? => ? = 5 - 1
[4,4,1,1,1]
 => [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
 => [8,4,3,2,9,10,11,1,5,6,7] => ? => ? = 3 - 1
[3,3,3,1,1]
 => [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
 => [9,6,3,10,11,2,7,8,1,4,5] => ? => ? = 4 - 1
[2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
 => ? => ? => ? = 6 - 1
[7,4,1]
 => [[1,3,4,5,10,11,12],[2,7,8,9],[6]]
 => ? => ? => ? = 5 - 1
[6,2,2,1,1]
 => [[1,4,9,10,11,12],[2,6],[3,8],[5],[7]]
 => ? => ? => ? = 8 - 1
[5,5,1,1]
 => [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
 => [8,3,2,9,10,11,12,1,4,5,6,7] => ? => ? = 3 - 1
[5,3,3,1]
 => [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
 => [8,5,9,10,2,6,7,1,3,4,11,12] => ? => ? = 4 - 1
[4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
 => ? => ? => ? = 4 - 1
[4,4,1,1,1,1]
 => [[1,6,7,8],[2,10,11,12],[3],[4],[5],[9]]
 => ? => ? => ? = 5 - 1
[3,3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
 => ? => ? => ? = 7 - 1
[3,3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3,11,12],[4],[7],[10]]
 => ? => ? => ? = 7 - 1
[6,2,2,2,1]
 => [[1,3,10,11,12,13],[2,5],[4,7],[6,9],[8]]
 => ? => ? => ? = 8 - 1
[4,4,2,2,1]
 => [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
 => [10,6,11,4,7,2,5,12,13,1,3,8,9] => ? => ? = 5 - 1
[3,3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
 => ? => ? => ? = 6 - 1
[7,5,1,1]
 => [[1,4,5,6,7,13,14],[2,9,10,11,12],[3],[8]]
 => ? => ? => ? = 3 - 1
[6,4,4]
 => [[1,2,3,4,13,14],[5,6,7,8],[9,10,11,12]]
 => ? => ? => ? = 5 - 1
[6,3,3,1,1]
 => [[1,4,5,12,13,14],[2,7,8],[3,10,11],[6],[9]]
 => ? => ? => ? = 6 - 1
[6,2,2,2,2]
 => [[1,2,11,12,13,14],[3,4],[5,6],[7,8],[9,10]]
 => ? => ? => ? = 6 - 1
[5,5,2,2]
 => [[1,2,7,8,9],[3,4,12,13,14],[5,6],[10,11]]
 => ? => ? => ? = 7 - 1
[5,5,1,1,1,1]
 => [[1,6,7,8,9],[2,11,12,13,14],[3],[4],[5],[10]]
 => ? => ? => ? = 3 - 1
[5,3,3,3]
 => [[1,2,3,13,14],[4,5,6],[7,8,9],[10,11,12]]
 => ? => ? => ? = 7 - 1
Description
The number of augmented double ascents of a permutation. An augmented double ascent of a permutation $\pi$ is a double ascent of the augmented permutation $\tilde\pi$ obtained from $\pi$ by adding an initial $0$. A double ascent of $\tilde\pi$ then is a position $i$ such that $\tilde\pi(i) < \tilde\pi(i+1) < \tilde\pi(i+2)$.
Matching statistic: St000735
Mapping
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000735: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 20%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 4 = 1 + 3
[2]
 => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 5 = 2 + 3
[1,1]
 => [2]
 => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 4 = 1 + 3
[1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 4 = 1 + 3
[3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => 5 = 2 + 3
[1,1,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 4 = 1 + 3
[2,2,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,9],[4,6,7,8,10]]
 => ? = 3 + 3
[1,1,1,1,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 4 = 1 + 3
[4,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[1,3,4,5,6,8],[2,7,9,10,11,12]]
 => ? = 2 + 3
[3,3]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
 => ? = 3 + 3
[2,2,2]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 5 + 3
[2,2,1,1]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[1,2,3,6,9],[4,5,7,8,10]]
 => ? = 5 + 3
[1,1,1,1,1,1]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 1 + 3
[2,2,2,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 4 + 3
[2,2,1,1,1]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => ? = 7 + 3
[5,3]
 => [5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[1,2,3,4,8,11],[5,6,7,9,10,12]]
 => ? = 3 + 3
[5,1,1,1]
 => [5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,7,11],[4,6,8,9,10,12]]
 => ? = 2 + 3
[4,2,2]
 => [4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [[1,3,4,5,6,10],[2,7,8,9,11,12]]
 => ? = 4 + 3
[3,3,1,1]
 => [6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,5,8,13],[6,7,9,10,11,12,14]]
 => ? = 3 + 3
[2,2,2,2]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,9,10],[5,6,7,8,11,12]]
 => ? = 6 + 3
[2,2,2,1,1]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => ? = 7 + 3
[2,2,1,1,1,1]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => ? = 9 + 3
[6,3]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[1,2,3,7,8,9],[4,5,6,10,11,12]]
 => ? = 5 + 3
[4,4,1]
 => [8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,7,9,17],[8,10,11,12,13,14,15,16,18]]
 => ? = 3 + 3
[3,3,1,1,1]
 => [6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,6,8,13],[5,7,9,10,11,12,14]]
 => ? = 5 + 3
[2,2,2,2,1]
 => [4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [[1,2,3,5,9,10],[4,6,7,8,11,12]]
 => ? = 5 + 3
[2,2,2,1,1,1]
 => [4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [[1,3,4,5,7,10],[2,6,8,9,11,12]]
 => ? = 10 + 3
[7,3]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,10,15],[7,8,9,11,12,13,14,16]]
 => ? = 7 + 3
[6,1,1,1,1]
 => [3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [[1,2,5,6,8,9],[3,4,7,10,11,12]]
 => ? = 2 + 3
[5,2,2,1]
 => [5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [[1,2,3,5,9,11],[4,6,7,8,10,12]]
 => ? = 6 + 3
[3,3,3,1]
 => [6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [[1,2,3,4,6,9,13],[5,7,8,10,11,12,14]]
 => ? = 5 + 3
[3,3,1,1,1,1]
 => [6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [[1,2,3,4,7,8,13],[5,6,9,10,11,12,14]]
 => ? = 7 + 3
[2,2,2,2,2]
 => [4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [[1,2,4,5,9,10],[3,6,7,8,11,12]]
 => ? = 7 + 3
[2,2,2,2,1,1]
 => [4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [[1,2,3,6,9,10],[4,5,7,8,11,12]]
 => ? = 9 + 3
[6,4,1]
 => [3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [[1,3,4,5,6,7,10,11],[2,8,9,12,13,14,15,16]]
 => ? = 3 + 3
[5,2,2,2]
 => [5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [[1,2,4,5,9,11],[3,6,7,8,10,12]]
 => ? = 5 + 3
[4,4,1,1,1]
 => [8,2,1]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,8,10,17],[7,9,11,12,13,14,15,16,18]]
 => ? = 3 + 3
[3,3,3,1,1]
 => [6,3,2]
 => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [[1,2,3,4,7,9,13],[5,6,8,10,11,12,14]]
 => ? = 4 + 3
[2,2,2,2,2,1]
 => [4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[1,3,4,5,9,10],[2,6,7,8,11,12]]
 => ? = 6 + 3
[7,4,1]
 => [7,1,1,1,1,1]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[1,2,4,5,6,7,8,15],[3,9,10,11,12,13,14,16]]
 => ? = 5 + 3
[6,2,2,1,1]
 => [4,3,3,2]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [[1,2,5,7,8,10],[3,4,6,9,11,12]]
 => ? = 8 + 3
[5,5,1,1]
 => [10,2]
 => [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
 => ?
 => ? = 3 + 3
[5,3,3,1]
 => [6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [[1,2,3,4,6,11,13],[5,7,8,9,10,12,14]]
 => ? = 4 + 3
[4,4,4]
 => [8,1,1,1,1]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,6,7,8,9,17],[5,10,11,12,13,14,15,16,18]]
 => ? = 4 + 3
[4,4,1,1,1,1]
 => [8,2,2]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,9,10,17],[7,8,11,12,13,14,15,16,18]]
 => ? = 5 + 3
[3,3,3,3]
 => [6,6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,5,6,13,14],[7,8,9,10,11,12,15,16]]
 => ? = 7 + 3
[3,3,3,1,1,1]
 => [6,3,2,1]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [[1,2,3,5,7,9,13],[4,6,8,10,11,12,14]]
 => ? = 7 + 3
[6,2,2,2,1]
 => [4,3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [[1,3,4,5,8,9,11],[2,6,7,10,12,13,14]]
 => ? = 8 + 3
[4,4,2,2,1]
 => [8,4,1]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,8,12,17],[7,9,10,11,13,14,15,16,18]]
 => ? = 5 + 3
[3,3,3,3,1]
 => [6,6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
 => [[1,2,3,4,5,7,13,14],[6,8,9,10,11,12,15,16]]
 => ? = 6 + 3
[7,5,1,1]
 => [7,5,2]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0]
 => [[1,2,3,4,5,8,12,15],[6,7,9,10,11,13,14,16]]
 => ? = 3 + 3
[6,4,4]
 => [8,3,3]
 => [1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,1,0]
 => [[1,2,3,4,5,6,10,11,17],[7,8,9,12,13,14,15,16,18]]
 => ? = 5 + 3
[6,3,3,1,1]
 => [6,3,3,2]
 => [1,1,1,0,0,1,0,1,1,0,0,0,1,0]
 => [[1,2,3,6,8,9,13],[4,5,7,10,11,12,14]]
 => ? = 6 + 3
[6,2,2,2,2]
 => [4,4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [[1,2,3,7,8,10,11],[4,5,6,9,12,13,14]]
 => ? = 6 + 3
[5,5,2,2]
 => [10,4]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0,1,0]
 => ?
 => ? = 7 + 3
[5,5,1,1,1,1]
 => [10,2,2]
 => [1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,1,0]
 => ?
 => ? = 3 + 3
[5,3,3,3]
 => [6,5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [[1,2,3,4,8,11,13],[5,6,7,9,10,12,14]]
 => ? = 7 + 3
Description
The last entry on the main diagonal of a standard tableau.