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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000159
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2] => [2,1]
=> [1]
=> []
=> 0
[2,1,3] => [2,1]
=> [1]
=> []
=> 0
[3,2,1] => [2,1]
=> [1]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> []
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> []
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> []
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> 0
[4,1,3,2] => [3,1]
=> [1]
=> []
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> []
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> []
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,3,2] => [2,1]
=> [1]
=> []
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [1]
=> []
=> 1 = 0 + 1
[3,2,1] => [2,1]
=> [1]
=> []
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1]
=> []
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> 1 = 0 + 1
[2,3,1,4] => [3,1]
=> [1]
=> []
=> 1 = 0 + 1
[2,4,3,1] => [3,1]
=> [1]
=> []
=> 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> [2]
=> []
=> 1 = 0 + 1
[4,1,3,2] => [3,1]
=> [1]
=> []
=> 1 = 0 + 1
[4,2,1,3] => [3,1]
=> [1]
=> []
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[4,3,2,1] => [2,2]
=> [2]
=> []
=> 1 = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 2 = 1 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 2 = 1 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> 1 = 0 + 1
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> 1 = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> 1 = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 2 = 1 + 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> 1 = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> 1 = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 2 = 1 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> 1 = 0 + 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 2 = 1 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 2 = 1 + 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 2 = 1 + 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 2 = 1 + 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000340
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> [1,0]
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,2] => [2,1]
=> [1]
=> [1,0]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1,0]
=> 0
[3,2,1] => [2,1]
=> [1]
=> [1,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,4,2] => [3,1]
=> [1]
=> [1,0]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,4,3] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> [1,0]
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> [1,0]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[4,1,3,2] => [3,1]
=> [1]
=> [1,0]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> [1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,3,2,1] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,4,5,6,7,8,10,9] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[2,1,3,4,5,6,7,8,9,10,11] => [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,5,6,7,8,9,10] => [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,2,1,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[11,12,3,4,5,6,10,9,8,7,1,2] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,10,12,4,5,6,11,9,8,2,7,3] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,2,9,12,5,6,11,10,3,8,7,4] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,2,3,8,12,6,11,4,10,9,7,5] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,2,3,4,7,12,5,11,10,9,8,6] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,2,3,4,5,6,12,11,10,9,8,7] => [2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[11,12,9,10,5,6,7,8,3,4,1,2] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[11,12,3,8,9,6,7,4,5,10,1,2] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[11,12,3,4,7,8,5,6,9,10,1,2] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[11,12,3,4,5,6,7,8,9,10,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,10,11,4,5,6,7,8,9,12,1,2] => [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,10,11,12,5,6,7,8,1,2,3,4] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,10,11,8,9,6,7,4,5,2,3,12] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,10,11,4,7,8,5,6,9,2,3,12] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[11,12,1,4,5,6,7,8,9,2,3,10] => [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,10,11,4,5,6,7,8,9,2,3,12] => [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,4,9,10,5,6,7,8,11,2,3,12] => [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,8,9,10,11,6,7,2,3,4,5,12] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,2,9,10,7,8,5,6,3,4,11,12] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,10,11,2,5,6,7,8,3,4,9,12] => [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,9,10,5,6,7,8,3,4,11,12] => [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,5,8,9,6,7,10,3,4,11,12] => [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,7,8,9,10,3,4,5,6,11,12] => [2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,2,9,10,3,6,7,4,5,8,11,12] => [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,8,9,6,7,4,5,10,11,12] => [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,6,7,8,9,4,5,10,11,12] => [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,8,9,4,5,6,7,10,11,12] => [3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,7,8,5,6,9,10,11,12] => [2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,5,6,7,8,9,10,11,12] => [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,5,6,7,10,9,8] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,4,3,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[9,2,3,4,5,6,7,8,1,10] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,10,3,4,5,6,7,8,9,2] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,5,6,10,8,9,7] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,5,10,7,8,9,6] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,10,6,7,8,9,5] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,10,5,6,7,8,9,4] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,10,4,5,6,7,8,9,3] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[10,2,3,4,5,6,7,8,9,1] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,5,6,7,8,10,11,9] => [3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,2,3,4,5,6,7,9,8,10] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[1,3,2,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[11,2,3,4,5,6,7,8,9,10,1] => [2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
Description
The number of non-final maximal constant sub-paths of length greater than one.
This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St001124
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,1]
=> [1]
=> []
=> ? = 0 - 1
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,3,2] => [2,1]
=> [1]
=> []
=> ? = 0 - 1
[2,1,3] => [2,1]
=> [1]
=> []
=> ? = 0 - 1
[3,2,1] => [2,1]
=> [1]
=> []
=> ? = 0 - 1
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? = 0 - 1
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? = 0 - 1
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? = 0 - 1
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? = 0 - 1
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? = 0 - 1
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? = 0 - 1
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? = 0 - 1
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? = 0 - 1
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? = 0 - 1
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? = 0 - 1
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? = 0 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> ? = 1 - 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? = 0 - 1
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? = 0 - 1
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? = 0 - 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> ? = 1 - 1
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? = 0 - 1
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? = 0 - 1
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? = 0 - 1
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> ? = 1 - 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> ? = 1 - 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> ? = 1 - 1
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? = 0 - 1
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? = 0 - 1
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> ? = 1 - 1
[2,3,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? = 0 - 1
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? = 0 - 1
[2,3,5,4,1] => [4,1]
=> [1]
=> []
=> ? = 0 - 1
[2,4,1,3,5] => [4,1]
=> [1]
=> []
=> ? = 0 - 1
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? = 1 - 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0 = 1 - 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,3,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,5,4,6,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,5,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,6,3,5,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,6,4,3,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,2,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,2,6,5,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,4,2,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,5,4,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,3,6,4,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,2,3,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,2,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,4,3,2,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,5,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,3,6,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,5,2,3,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,4,6,2,5,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,2,4,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,3,2,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,3,4,2,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[1,5,3,4,6,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,3,6,2,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
[1,5,4,3,2,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0 = 1 - 1
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St001200
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 50%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [1,2] => [1,0,1,0]
=> 2 = 0 + 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3 = 1 + 2
[1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 0 + 2
[3,2,1] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[1,4,3,2] => [1,3,4,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[2,4,3,1] => [3,4,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2 = 0 + 2
[3,2,1,4] => [2,3,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[3,2,4,1] => [2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2 = 0 + 2
[3,4,1,2] => [3,1,4,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[4,1,3,2] => [3,4,2,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[4,2,1,3] => [2,4,3,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
[4,2,3,1] => [2,3,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[4,3,2,1] => [3,2,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,3,5,2,4] => [1,5,4,2,3] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,3,5,4,2] => [1,4,5,2,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3 = 1 + 2
[1,4,2,5,3] => [1,5,3,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,4,3,5,2] => [1,3,5,2,4] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3 = 1 + 2
[1,4,5,2,3] => [1,4,2,5,3] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[1,4,5,3,2] => [1,5,2,4,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,5,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,5,2,4,3] => [1,4,5,3,2] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,5,3,2,4] => [1,3,5,4,2] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3 = 1 + 2
[1,5,3,4,2] => [1,3,4,5,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[1,5,4,2,3] => [1,5,3,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
[1,5,4,3,2] => [1,4,3,5,2] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3 = 1 + 2
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 1 + 2
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 1 + 2
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 1 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 2
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1 + 2
[1,2,3,5,6,4] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 2
[1,2,4,5,3,6] => [1,2,5,3,4,6] => [1,2,5,3,4,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1 + 2
[1,2,4,5,6,3] => [1,2,6,3,4,5] => [1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,2,4,6,3,5] => [1,2,6,5,3,4] => [1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,2,4,6,5,3] => [1,2,5,6,3,4] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 1 + 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1 + 2
[1,2,5,3,6,4] => [1,2,6,4,3,5] => [1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1 + 2
[1,2,5,4,6,3] => [1,2,4,6,3,5] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 1 + 2
[1,2,5,6,3,4] => [1,2,5,3,6,4] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 2
[1,2,5,6,4,3] => [1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,2,6,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,2,6,3,5,4] => [1,2,5,6,4,3] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,6,4,3,5] => [1,2,4,6,5,3] => [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,2,6,4,5,3] => [1,2,4,5,6,3] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1 + 2
[1,2,6,5,3,4] => [1,2,6,4,5,3] => [1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,2,4,3,6,5] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1 + 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1 + 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1 + 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 2
[1,3,2,5,6,4] => [1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 2
[1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,3,2,4,6,5] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1 + 2
[1,3,4,2,5,6] => [1,4,2,3,5,6] => [1,4,2,3,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 2
[1,3,4,2,6,5] => [1,4,2,3,6,5] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[1,3,4,5,2,6] => [1,5,2,3,4,6] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 2
[1,3,4,5,6,2] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,3,4,6,2,5] => [1,6,5,2,3,4] => [1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,3,4,6,5,2] => [1,5,6,2,3,4] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1 + 2
[1,3,5,2,4,6] => [1,5,4,2,3,6] => [1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 1 + 2
[1,3,5,2,6,4] => [1,6,4,2,3,5] => [1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,3,5,4,2,6] => [1,4,5,2,3,6] => [1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1 + 2
[1,3,5,4,6,2] => [1,4,6,2,3,5] => [1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1 + 2
[1,3,5,6,2,4] => [1,5,2,3,6,4] => [1,4,2,3,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
[1,3,5,6,4,2] => [1,6,2,3,5,4] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,3,6,2,4,5] => [1,6,5,4,2,3] => [1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,3,6,2,5,4] => [1,5,6,4,2,3] => [1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,3,6,4,2,5] => [1,4,6,5,2,3] => [1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 1 + 2
[1,3,6,4,5,2] => [1,4,5,6,2,3] => [1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 1 + 2
[1,3,6,5,2,4] => [1,6,4,5,2,3] => [1,6,3,5,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 2
[1,3,6,5,4,2] => [1,5,4,6,2,3] => [1,4,3,6,2,5] => [1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1 + 2
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 $eA$.
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