Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000159
Mp00108: Permutations cycle typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger 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 typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger 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 typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck 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 typeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
Mp00202: Integer partitions first row removalInteger 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.
Mp00087: Permutations inverse first fundamental transformationPermutations
Mp00254: Permutations Inverse fireworks mapPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck 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 $eA$.