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Your data matches 138 different statistics following compositions of up to 3 maps.
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Matching statistic: St000459
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 4
[2,2]
=> 3
[2,1,1]
=> 4
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 5
[3,2]
=> 4
[3,1,1]
=> 5
[2,2,1]
=> 4
[2,1,1,1]
=> 5
[1,1,1,1,1]
=> 5
[6]
=> 6
[5,1]
=> 6
[4,2]
=> 5
[4,1,1]
=> 6
[3,3]
=> 4
[3,2,1]
=> 5
[3,1,1,1]
=> 6
[2,2,2]
=> 4
[2,2,1,1]
=> 5
[2,1,1,1,1]
=> 6
[1,1,1,1,1,1]
=> 6
[7]
=> 7
[6,1]
=> 7
[5,2]
=> 6
[5,1,1]
=> 7
[4,3]
=> 5
[4,2,1]
=> 6
[4,1,1,1]
=> 7
[3,3,1]
=> 5
[3,2,2]
=> 5
[3,2,1,1]
=> 6
[3,1,1,1,1]
=> 7
[2,2,2,1]
=> 5
[2,2,1,1,1]
=> 6
[2,1,1,1,1,1]
=> 7
[1,1,1,1,1,1,1]
=> 7
[8]
=> 8
[7,1]
=> 8
[6,2]
=> 7
[6,1,1]
=> 8
[5,3]
=> 6
[5,2,1]
=> 7
Description
The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000460
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1] => [1]
=> 1
[2]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 2
[1,1]
=> [1,1,0,0]
=> [2] => [2]
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [3] => [3]
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 4
[2,2]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 5
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 4
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> 5
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 4
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [2,1,1,1,1]
=> 6
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 5
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> 6
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> 5
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 7
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,3] => [3,1,1,1,1]
=> 7
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 5
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> 6
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,4] => [4,1,1,1]
=> 7
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> 5
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 5
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> 6
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,5] => [5,1,1]
=> 7
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> 5
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> 6
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,6] => [6,1]
=> 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7] => [7]
=> 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 8
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,3] => [3,1,1,1,1]
=> 7
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => [3,1,1,1,1,1]
=> 8
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [4,1,1]
=> 6
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,4] => [4,1,1,1]
=> 7
[8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,4] => ?
=> ? = 11
[7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,5] => ?
=> ? = 11
[6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,7] => ?
=> ? = 11
[4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9
[4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,8] => ?
=> ? = 11
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9
[12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 12
[11,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 12
[10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,3] => ?
=> ? = 12
[9,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? => ?
=> ? = 11
[9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[8,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[8,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[7,2,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 10
[7,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 9
[6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[6,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[5,2,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[5,2,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[5,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9
[4,2,2,1,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[4,2,1,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 13
[12,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 13
[10,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? => ?
=> ? = 11
[9,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 11
[8,4,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? => ?
=> ? = 10
[8,3,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[7,5,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? => ?
=> ? = 9
[6,5,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 9
[6,4,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 9
[6,3,2,2]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 9
[6,3,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[6,2,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[6,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 13
[5,4,1,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[5,3,2,1,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1] => [1]
=> 1
[2]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 2
[1,1]
=> [1,1,0,0]
=> [2] => [2]
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [3] => [3]
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 4
[2,2]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 5
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 4
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> 5
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 4
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [2,1,1,1,1]
=> 6
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 5
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> 6
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> 5
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 7
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,3] => [3,1,1,1,1]
=> 7
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 5
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> 6
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,4] => [4,1,1,1]
=> 7
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> 5
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 5
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> 6
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,5] => [5,1,1]
=> 7
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> 5
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> 6
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,6] => [6,1]
=> 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7] => [7]
=> 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 8
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,3] => [3,1,1,1,1]
=> 7
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => [3,1,1,1,1,1]
=> 8
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [4,1,1]
=> 6
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,4] => [4,1,1,1]
=> 7
[8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,4] => ?
=> ? = 11
[7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,5] => ?
=> ? = 11
[6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,7] => ?
=> ? = 11
[4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9
[4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,8] => ?
=> ? = 11
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9
[12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 12
[11,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 12
[10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,3] => ?
=> ? = 12
[9,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? => ?
=> ? = 11
[9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[8,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[8,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[7,2,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 10
[7,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 9
[6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[6,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[5,2,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[5,2,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[5,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9
[4,2,2,1,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[4,2,1,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 13
[12,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 13
[10,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? => ?
=> ? = 11
[9,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 11
[8,4,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? => ?
=> ? = 10
[8,3,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[7,5,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? => ?
=> ? = 9
[6,5,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 9
[6,4,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 9
[6,3,2,2]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 9
[6,3,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[6,2,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[6,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 13
[5,4,1,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[5,3,2,1,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
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: St001382
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 72%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1] => [1]
=> 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 1 = 2 - 1
[1,1]
=> [1,1,0,0]
=> [2] => [2]
=> 1 = 2 - 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 2 = 3 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 2 = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [3] => [3]
=> 2 = 3 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 3 = 4 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 3 = 4 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 2 = 3 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 3 = 4 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 3 = 4 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 4 = 5 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 4 = 5 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 3 = 4 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> 4 = 5 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 3 = 4 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 4 = 5 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> 4 = 5 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 5 = 6 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [2,1,1,1,1]
=> 5 = 6 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 4 = 5 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> 5 = 6 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 3 = 4 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 4 = 5 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> 5 = 6 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 3 = 4 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> 4 = 5 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> 5 = 6 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> 5 = 6 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 6 = 7 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 6 = 7 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> 5 = 6 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,3] => [3,1,1,1,1]
=> 6 = 7 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 4 = 5 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> 5 = 6 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,4] => [4,1,1,1]
=> 6 = 7 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> 4 = 5 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 4 = 5 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> 5 = 6 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,5] => [5,1,1]
=> 6 = 7 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> 4 = 5 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,6] => [6,1]
=> 6 = 7 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7] => [7]
=> 6 = 7 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 7 = 8 - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 7 = 8 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,3] => [3,1,1,1,1]
=> 6 = 7 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => [3,1,1,1,1,1]
=> 7 = 8 - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [4,1,1]
=> 5 = 6 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,4] => [4,1,1,1]
=> 6 = 7 - 1
[8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,4] => ?
=> ? = 11 - 1
[7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
[7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,5] => ?
=> ? = 11 - 1
[6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
[5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
[5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,7] => ?
=> ? = 11 - 1
[4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9 - 1
[4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,8] => ?
=> ? = 11 - 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9 - 1
[12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 12 - 1
[11,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 12 - 1
[10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,3] => ?
=> ? = 12 - 1
[9,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? => ?
=> ? = 11 - 1
[9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12 - 1
[8,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 11 - 1
[8,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12 - 1
[7,2,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
[7,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11 - 1
[7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12 - 1
[6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 9 - 1
[6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
[6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11 - 1
[6,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12 - 1
[5,2,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
[5,2,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11 - 1
[5,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12 - 1
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9 - 1
[4,2,2,1,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
[4,2,1,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11 - 1
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12 - 1
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11 - 1
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12 - 1
[13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 13 - 1
[12,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 13 - 1
[10,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? => ?
=> ? = 11 - 1
[9,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 11 - 1
[8,4,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
[8,3,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 11 - 1
[7,5,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? => ?
=> ? = 9 - 1
[6,5,1,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 9 - 1
[6,4,2,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 9 - 1
[6,3,2,2]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? => ?
=> ? = 9 - 1
[6,3,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11 - 1
[6,2,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11 - 1
[6,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 13 - 1
[5,4,1,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
[5,3,2,1,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10 - 1
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
Matching statistic: St000228
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 71%●distinct values known / distinct values provided: 56%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 71%●distinct values known / distinct values provided: 56%
Values
[1]
=> [1,0]
=> [1] => [1]
=> 1
[2]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 2
[1,1]
=> [1,1,0,0]
=> [2] => [2]
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 4
[2,2]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 5
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 4
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 5
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 4
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [2,1,1,1,1]
=> 6
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 5
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,2,1] => [2,1,1,1,1]
=> 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,2,1,1] => [2,1,1,1,1]
=> 6
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => [3,1,1]
=> 5
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1] => [2,1,1,1,1]
=> 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1] => [2,1,1,1,1]
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 7
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,2,1] => [2,1,1,1,1,1]
=> 7
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 5
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,3,1] => [3,1,1,1]
=> 6
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1] => [2,1,1,1,1,1]
=> 7
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => [3,1,1]
=> 5
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 5
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,1,1] => [3,1,1,1]
=> 6
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1] => [2,1,1,1,1,1]
=> 7
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => [4,1]
=> 5
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,1,1] => [3,1,1,1]
=> 6
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1] => [2,1,1,1,1,1]
=> 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1] => [2,1,1,1,1,1]
=> 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 8
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,3] => [3,1,1,1,1]
=> 7
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> 8
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,3,1] => [3,1,1,1]
=> 6
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,3,1] => [3,1,1,1,1]
=> 7
[11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 11
[10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 11
[9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,2,1] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 11
[8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,2,1,1] => ?
=> ? = 11
[7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,2,1,1,1] => ?
=> ? = 11
[6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[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]
=> [1,1,1,1,1,2,1,1,1,1] => ?
=> ? = 11
[5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,2,1,1,1,1,1] => ?
=> ? = 11
[4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9
[4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,2,1,1,1,1,1,1] => ?
=> ? = 11
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1,1,1,1,1] => ?
=> ? = 11
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1,1,1,1] => ?
=> ? = 11
[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]
=> [2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 11
[12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 12
[11,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 12
[10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => [3,1,1,1,1,1,1,1,1]
=> ? = 11
[10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? => ?
=> ? = 12
[9,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? => ?
=> ? = 11
[9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[8,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[8,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[7,2,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 10
[7,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? => ?
=> ? = 9
[6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[6,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[5,2,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[5,2,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[5,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 9
[4,2,2,1,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[4,2,1,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[4,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 10
[3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 11
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[2,2,1,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,1,1,1,1,1,1,1] => ?
=> ? = 11
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1,1]
=> ? = 12
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ?
=> ? = 12
[13]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ?
=> ? = 13
[12,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ?
=> ? = 13
[10,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,3,1] => [3,1,1,1,1,1,1,1,1]
=> ? = 11
[9,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? => ?
=> ? = 11
[8,4,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? => ?
=> ? = 10
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001437
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 61% ●values known / values provided: 66%●distinct values known / distinct values provided: 61%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 61% ●values known / values provided: 66%●distinct values known / distinct values provided: 61%
Values
[1]
=> [1,0]
=> [1] => 1 => 1
[2]
=> [1,0,1,0]
=> [1,1] => 11 => 2
[1,1]
=> [1,1,0,0]
=> [2] => 10 => 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [3] => 100 => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 4
[2,2]
=> [1,1,1,0,0,0]
=> [3] => 100 => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1000 => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 5
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 4
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 5
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 4
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => 111111 => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 111110 => 6
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 5
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => 111100 => 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1000 => 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => 111000 => 6
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 5
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => 110000 => 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => 100000 => 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => 1111111 => 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => 1111110 => 7
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => 111100 => 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,3] => 1111100 => 7
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 5
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => 111000 => 6
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,4] => 1111000 => 7
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 10000 => 5
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 5
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => 110000 => 6
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,5] => 1110000 => 7
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 5
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => 100000 => 6
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,6] => 1100000 => 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7] => 1000000 => 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => 11111111 => 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => 11111110 => 8
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,3] => 1111100 => 7
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => 11111100 => 8
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => 111000 => 6
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,4] => 1111000 => 7
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => 1111111100 => ? = 10
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[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]
=> [1,1,1,1,1,5] => 1111110000 => ? = 10
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,7] => 1111000000 => ? = 10
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,8] => 1110000000 => ? = 10
[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]
=> [1,9] => 1100000000 => ? = 10
[11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => 11111111111 => ? = 11
[9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => 1111111100 => ? = 10
[9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => 11111111100 => ? = 11
[8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,4] => ? => ? = 11
[7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ? => ? = 10
[7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,5] => ? => ? = 11
[6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 10
[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]
=> [1,1,1,1,1,6] => 11111100000 => ? = 11
[5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 10
[5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,7] => ? => ? = 11
[4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 9
[4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 10
[4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,8] => ? => ? = 11
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 9
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,9] => 1100000000 => ? = 10
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,9] => 11100000000 => ? = 11
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,10] => 11000000000 => ? = 11
[12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ? => ? = 12
[11,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ? => ? = 12
[10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => 11111111100 => ? = 11
[10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,3] => ? => ? = 12
[9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[9,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? => ? => ? = 11
[9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? => ? => ? = 12
[8,3,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,5] => 1111110000 => ? = 10
[8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[8,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ? => ? = 11
[8,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 12
[7,2,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? => ? => ? = 10
[7,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 11
[7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 12
[6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? => ? => ? = 9
[6,3,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,7] => 1111000000 => ? = 10
[6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? => ? => ? = 10
[6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 11
[6,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 12
[5,3,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,8] => 1110000000 => ? = 10
[5,2,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 10
[5,2,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 11
[5,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 12
[4,3,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 9
[4,3,1,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,9] => 1100000000 => ? = 10
Description
The flex of a binary word.
This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].
Matching statistic: St000393
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000393: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0]
=> [1] => 1 => 1
[2]
=> [1,0,1,0]
=> [1,1] => 11 => 2
[1,1]
=> [1,1,0,0]
=> [2] => 10 => 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [3] => 100 => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 4
[2,2]
=> [1,1,1,0,0,0]
=> [3] => 100 => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1000 => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 5
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 4
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 5
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 4
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => 111111 => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 111110 => 6
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 5
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => 111100 => 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1000 => 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => 111000 => 6
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 5
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => 110000 => 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => 100000 => 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => 1111111 => 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => 1111110 => 7
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => 111100 => 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,3] => 1111100 => 7
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 5
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => 111000 => 6
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,4] => 1111000 => 7
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 10000 => 5
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 5
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => 110000 => 6
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,5] => 1110000 => 7
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 5
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => 100000 => 6
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,6] => 1100000 => 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7] => 1000000 => 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => 11111111 => 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => 11111110 => 8
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,3] => 1111100 => 7
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => 11111100 => 8
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => 111000 => 6
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,4] => 1111000 => 7
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => 1111111110 => ? = 10
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => 1111111100 => ? = 10
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[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]
=> [1,1,1,1,1,5] => 1111110000 => ? = 10
[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]
=> [1,1,1,1,6] => 1111100000 => ? = 10
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,7] => 1111000000 => ? = 10
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,8] => 1110000000 => ? = 10
[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]
=> [1,9] => 1100000000 => ? = 10
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10] => 1000000000 => ? = 10
[11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => 11111111111 => ? = 11
[10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => 11111111110 => ? = 11
[9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => 1111111100 => ? = 10
[9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => 11111111100 => ? = 11
[8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,4] => ? => ? = 11
[7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ? => ? = 10
[7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,5] => ? => ? = 11
[6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 10
[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]
=> [1,1,1,1,1,6] => 11111100000 => ? = 11
[5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 10
[5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,7] => ? => ? = 11
[4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 9
[4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 10
[4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,8] => ? => ? = 11
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 9
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,9] => 1100000000 => ? = 10
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,9] => 11100000000 => ? = 11
[2,2,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10] => 1000000000 => ? = 10
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,10] => 11000000000 => ? = 11
[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]
=> [11] => 10000000000 => ? = 11
[12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ? => ? = 12
[11,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ? => ? = 12
[10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => 11111111100 => ? = 11
[10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,3] => ? => ? = 12
[9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[9,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? => ? => ? = 11
[9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? => ? => ? = 12
[8,3,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,5] => 1111110000 => ? = 10
[8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[8,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ? => ? = 11
[8,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 12
[7,3,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,6] => 1111100000 => ? = 10
[7,2,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? => ? => ? = 10
[7,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 11
[7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 12
[6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? => ? => ? = 9
[6,3,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,7] => 1111000000 => ? = 10
[6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? => ? => ? = 10
[6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 11
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St001267
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001267: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001267: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0]
=> [1] => 1 => 1
[2]
=> [1,0,1,0]
=> [1,1] => 11 => 2
[1,1]
=> [1,1,0,0]
=> [2] => 10 => 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [3] => 100 => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 4
[2,2]
=> [1,1,1,0,0,0]
=> [3] => 100 => 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1000 => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 5
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 4
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 5
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 4
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => 111111 => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 111110 => 6
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 5
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => 111100 => 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1000 => 4
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => 111000 => 6
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 4
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 5
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => 110000 => 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => 100000 => 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => 1111111 => 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => 1111110 => 7
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => 111100 => 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,3] => 1111100 => 7
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 5
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => 111000 => 6
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,4] => 1111000 => 7
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 10000 => 5
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 5
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => 110000 => 6
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,5] => 1110000 => 7
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 5
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => 100000 => 6
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,6] => 1100000 => 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7] => 1000000 => 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => 11111111 => 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => 11111110 => 8
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,3] => 1111100 => 7
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => 11111100 => 8
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => 111000 => 6
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,4] => 1111000 => 7
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => 1111111110 => ? = 10
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => 1111111100 => ? = 10
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[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]
=> [1,1,1,1,1,5] => 1111110000 => ? = 10
[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]
=> [1,1,1,1,6] => 1111100000 => ? = 10
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,7] => 1111000000 => ? = 10
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,8] => 1110000000 => ? = 10
[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]
=> [1,9] => 1100000000 => ? = 10
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10] => 1000000000 => ? = 10
[11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => 11111111111 => ? = 11
[10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => 11111111110 => ? = 11
[9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => 1111111100 => ? = 10
[9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => 11111111100 => ? = 11
[8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,4] => ? => ? = 11
[7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ? => ? = 10
[7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,5] => ? => ? = 11
[6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 10
[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]
=> [1,1,1,1,1,6] => 11111100000 => ? = 11
[5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 10
[5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,7] => ? => ? = 11
[4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 9
[4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 10
[4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,8] => ? => ? = 11
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 9
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,9] => 1100000000 => ? = 10
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,9] => 11100000000 => ? = 11
[2,2,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10] => 1000000000 => ? = 10
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,10] => 11000000000 => ? = 11
[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]
=> [11] => 10000000000 => ? = 11
[12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ? => ? = 12
[11,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ? => ? = 12
[10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => 11111111100 => ? = 11
[10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,3] => ? => ? = 12
[9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[9,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? => ? => ? = 11
[9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? => ? => ? = 12
[8,3,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,5] => 1111110000 => ? = 10
[8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,4] => 1111111000 => ? = 10
[8,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ? => ? = 11
[8,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 12
[7,3,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,6] => 1111100000 => ? = 10
[7,2,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? => ? => ? = 10
[7,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 11
[7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 12
[6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? => ? => ? = 9
[6,3,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,7] => 1111000000 => ? = 10
[6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? => ? => ? = 10
[6,2,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? => ? = 11
Description
The length of the Lyndon factorization of the binary word.
The Lyndon factorization of a finite word w is its unique factorization as a non-increasing product of Lyndon words, i.e., $w = l_1\dots l_n$ where each $l_i$ is a Lyndon word and $l_1 \geq\dots\geq l_n$.
Matching statistic: St000806
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 44% ●values known / values provided: 62%●distinct values known / distinct values provided: 44%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 44% ●values known / values provided: 62%●distinct values known / distinct values provided: 44%
Values
[1]
=> [1,0]
=> [1] => ? = 1 + 1
[2]
=> [1,0,1,0]
=> [1,1] => 3 = 2 + 1
[1,1]
=> [1,1,0,0]
=> [2] => 3 = 2 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => 4 = 3 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => 4 = 3 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [3] => 4 = 3 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 5 = 4 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 5 = 4 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3] => 4 = 3 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 5 = 4 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [4] => 5 = 4 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 6 = 5 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 6 = 5 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 5 = 4 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 6 = 5 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [4] => 5 = 4 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 6 = 5 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 6 = 5 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => 7 = 6 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => 7 = 6 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 6 = 5 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => 7 = 6 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4] => 5 = 4 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 6 = 5 + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => 7 = 6 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => 5 = 4 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 6 = 5 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => 7 = 6 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => 7 = 6 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1] => 8 = 7 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => 8 = 7 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => 7 = 6 + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,3] => 8 = 7 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 6 = 5 + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => 7 = 6 + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,4] => 8 = 7 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 6 = 5 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 6 = 5 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => 7 = 6 + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,5] => 8 = 7 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 6 = 5 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => 7 = 6 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,6] => 8 = 7 + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7] => 8 = 7 + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => 9 = 8 + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,2] => 9 = 8 + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,3] => 8 = 7 + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,3] => 9 = 8 + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => 7 = 6 + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,4] => 8 = 7 + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,4] => 9 = 8 + 1
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => ? = 10 + 1
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,2] => ? = 10 + 1
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,3] => ? = 10 + 1
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,4] => ? = 10 + 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]
=> [1,1,1,1,1,5] => ? = 10 + 1
[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]
=> [1,1,1,1,6] => ? = 10 + 1
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,7] => ? = 10 + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,8] => ? = 10 + 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]
=> [1,9] => ? = 10 + 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10] => ? = 10 + 1
[11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1] => ? = 11 + 1
[10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,2] => ? = 11 + 1
[9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,3] => ? = 10 + 1
[9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,3] => ? = 11 + 1
[8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,4] => ? = 10 + 1
[8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,4] => ? = 11 + 1
[7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ? = 10 + 1
[7,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,5] => ? = 11 + 1
[6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ? = 10 + 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]
=> [1,1,1,1,1,6] => ? = 11 + 1
[5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 10 + 1
[5,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,7] => ? = 11 + 1
[4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? => ? = 9 + 1
[4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 10 + 1
[4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,8] => ? = 11 + 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 9 + 1
[3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,9] => ? = 10 + 1
[3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,9] => ? = 11 + 1
[2,2,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10] => ? = 10 + 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,10] => ? = 11 + 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]
=> [11] => ? = 11 + 1
[12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? => ? = 12 + 1
[11,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? => ? = 12 + 1
[10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,3] => ? = 11 + 1
[10,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,3] => ? = 12 + 1
[9,3]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,4] => ? = 10 + 1
[9,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? => ? = 11 + 1
[9,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? => ? = 12 + 1
[8,3,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,5] => ? = 10 + 1
[8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,4] => ? = 10 + 1
[8,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? => ? = 11 + 1
[8,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? => ? = 12 + 1
[7,3,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,6] => ? = 10 + 1
[7,2,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? => ? = 10 + 1
[7,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? => ? = 11 + 1
[7,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? => ? = 12 + 1
[6,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> ? => ? = 9 + 1
[6,3,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,7] => ? = 10 + 1
[6,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? => ? = 10 + 1
Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
Matching statistic: St000543
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 56%●distinct values known / distinct values provided: 50%
Mp00096: Binary words —Foata bijection⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000543: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 56%●distinct values known / distinct values provided: 50%
Values
[1]
=> 10 => 10 => 01 => 2 = 1 + 1
[2]
=> 100 => 010 => 001 => 3 = 2 + 1
[1,1]
=> 110 => 110 => 011 => 3 = 2 + 1
[3]
=> 1000 => 0010 => 0001 => 4 = 3 + 1
[2,1]
=> 1010 => 1100 => 0011 => 4 = 3 + 1
[1,1,1]
=> 1110 => 1110 => 0111 => 4 = 3 + 1
[4]
=> 10000 => 00010 => 00001 => 5 = 4 + 1
[3,1]
=> 10010 => 10100 => 00011 => 5 = 4 + 1
[2,2]
=> 1100 => 0110 => 0011 => 4 = 3 + 1
[2,1,1]
=> 10110 => 11010 => 00111 => 5 = 4 + 1
[1,1,1,1]
=> 11110 => 11110 => 01111 => 5 = 4 + 1
[5]
=> 100000 => 000010 => 000001 => 6 = 5 + 1
[4,1]
=> 100010 => 100100 => 000011 => 6 = 5 + 1
[3,2]
=> 10100 => 01100 => 00011 => 5 = 4 + 1
[3,1,1]
=> 100110 => 101010 => 001011 => 6 = 5 + 1
[2,2,1]
=> 11010 => 11100 => 00111 => 5 = 4 + 1
[2,1,1,1]
=> 101110 => 110110 => 001111 => 6 = 5 + 1
[1,1,1,1,1]
=> 111110 => 111110 => 011111 => 6 = 5 + 1
[6]
=> 1000000 => 0000010 => 0000001 => 7 = 6 + 1
[5,1]
=> 1000010 => 1000100 => 0000011 => 7 = 6 + 1
[4,2]
=> 100100 => 010100 => 000101 => 6 = 5 + 1
[4,1,1]
=> 1000110 => 1001010 => 0001011 => 7 = 6 + 1
[3,3]
=> 11000 => 00110 => 00011 => 5 = 4 + 1
[3,2,1]
=> 101010 => 111000 => 000111 => 6 = 5 + 1
[3,1,1,1]
=> 1001110 => 1010110 => 0010111 => 7 = 6 + 1
[2,2,2]
=> 11100 => 01110 => 00111 => 5 = 4 + 1
[2,2,1,1]
=> 110110 => 111010 => 001111 => 6 = 5 + 1
[2,1,1,1,1]
=> 1011110 => 1101110 => 0011111 => 7 = 6 + 1
[1,1,1,1,1,1]
=> 1111110 => 1111110 => 0111111 => 7 = 6 + 1
[7]
=> 10000000 => 00000010 => 00000001 => 8 = 7 + 1
[6,1]
=> 10000010 => 10000100 => 00000011 => 8 = 7 + 1
[5,2]
=> 1000100 => 0100100 => 0000101 => 7 = 6 + 1
[5,1,1]
=> 10000110 => 10001010 => 00001011 => 8 = 7 + 1
[4,3]
=> 101000 => 001100 => 000011 => 6 = 5 + 1
[4,2,1]
=> 1001010 => 1101000 => 0000111 => 7 = 6 + 1
[4,1,1,1]
=> 10001110 => 10010110 => 00010111 => 8 = 7 + 1
[3,3,1]
=> 110010 => 101100 => 000111 => 6 = 5 + 1
[3,2,2]
=> 101100 => 011010 => 001011 => 6 = 5 + 1
[3,2,1,1]
=> 1010110 => 1110010 => 0001111 => 7 = 6 + 1
[3,1,1,1,1]
=> 10011110 => 10101110 => 00101111 => 8 = 7 + 1
[2,2,2,1]
=> 111010 => 111100 => 001111 => 6 = 5 + 1
[2,2,1,1,1]
=> 1101110 => 1110110 => 0011111 => 7 = 6 + 1
[2,1,1,1,1,1]
=> 10111110 => 11011110 => 00111111 => 8 = 7 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 11111110 => 01111111 => 8 = 7 + 1
[8]
=> 100000000 => 000000010 => 000000001 => 9 = 8 + 1
[7,1]
=> 100000010 => 100000100 => 000000011 => 9 = 8 + 1
[6,2]
=> 10000100 => 01000100 => 00000101 => 8 = 7 + 1
[6,1,1]
=> 100000110 => 100001010 => 000001011 => 9 = 8 + 1
[5,3]
=> 1001000 => 0010100 => 0000101 => 7 = 6 + 1
[5,2,1]
=> 10001010 => 11001000 => 00000111 => 8 = 7 + 1
[3,1,1,1,1,1,1]
=> 1001111110 => 1010111110 => 0010111111 => ? = 9 + 1
[2,1,1,1,1,1,1,1]
=> 1011111110 => 1101111110 => 0011111111 => ? = 9 + 1
[1,1,1,1,1,1,1,1,1]
=> 1111111110 => 1111111110 => 0111111111 => ? = 9 + 1
[10]
=> 10000000000 => 00000000010 => 00000000001 => ? = 10 + 1
[9,1]
=> 10000000010 => 10000000100 => 00000000011 => ? = 10 + 1
[8,1,1]
=> 10000000110 => 10000001010 => 00000001011 => ? = 10 + 1
[7,1,1,1]
=> 10000001110 => 10000010110 => 00000010111 => ? = 10 + 1
[6,1,1,1,1]
=> 10000011110 => 10000101110 => 00000101111 => ? = 10 + 1
[5,1,1,1,1,1]
=> 10000111110 => 10001011110 => 00001011111 => ? = 10 + 1
[4,1,1,1,1,1,1]
=> 10001111110 => 10010111110 => 00010111111 => ? = 10 + 1
[3,1,1,1,1,1,1,1]
=> 10011111110 => 10101111110 => 00101111111 => ? = 10 + 1
[2,2,1,1,1,1,1,1]
=> 1101111110 => 1110111110 => 0011111111 => ? = 9 + 1
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => 11011111110 => 00111111111 => ? = 10 + 1
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 11111111110 => 01111111111 => ? = 10 + 1
[11]
=> 100000000000 => ? => ? => ? = 11 + 1
[10,1]
=> 100000000010 => ? => ? => ? = 11 + 1
[9,2]
=> 10000000100 => 01000000100 => ? => ? = 10 + 1
[9,1,1]
=> 100000000110 => ? => ? => ? = 11 + 1
[8,2,1]
=> 10000001010 => 11000001000 => ? => ? = 10 + 1
[8,1,1,1]
=> 100000001110 => ? => ? => ? = 11 + 1
[7,3,1]
=> 1000010010 => ? => ? => ? = 9 + 1
[7,2,2]
=> 1000001100 => ? => ? => ? = 9 + 1
[7,2,1,1]
=> 10000010110 => 11000010010 => ? => ? = 10 + 1
[7,1,1,1,1]
=> 100000011110 => ? => ? => ? = 11 + 1
[6,3,1,1]
=> 1000100110 => ? => ? => ? = 9 + 1
[6,2,2,1]
=> 1000011010 => ? => ? => ? = 9 + 1
[6,2,1,1,1]
=> 10000101110 => 11000100110 => ? => ? = 10 + 1
[6,1,1,1,1,1]
=> 100000111110 => 100001011110 => ? => ? = 11 + 1
[5,2,1,1,1,1]
=> 10001011110 => 11001001110 => ? => ? = 10 + 1
[5,1,1,1,1,1,1]
=> 100001111110 => ? => ? => ? = 11 + 1
[4,3,1,1,1,1]
=> 1010011110 => ? => ? => ? = 9 + 1
[4,2,2,1,1,1]
=> 1001101110 => ? => ? => ? = 9 + 1
[4,2,1,1,1,1,1]
=> 10010111110 => 11010011110 => ? => ? = 10 + 1
[4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? => ? = 11 + 1
[3,3,1,1,1,1,1]
=> 1100111110 => ? => ? => ? = 9 + 1
[3,2,2,1,1,1,1]
=> 1011011110 => ? => ? => ? = 9 + 1
[3,2,1,1,1,1,1,1]
=> 10101111110 => 11100111110 => ? => ? = 10 + 1
[3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? => ? = 11 + 1
[2,2,2,1,1,1,1,1]
=> 1110111110 => 1111011110 => ? => ? = 9 + 1
[2,2,1,1,1,1,1,1,1]
=> 11011111110 => 11101111110 => ? => ? = 10 + 1
[2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? => ? = 11 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? => ? = 11 + 1
[12]
=> 1000000000000 => ? => ? => ? = 12 + 1
[11,1]
=> 1000000000010 => ? => ? => ? = 12 + 1
[10,2]
=> 100000000100 => 010000000100 => ? => ? = 11 + 1
[10,1,1]
=> 1000000000110 => ? => ? => ? = 12 + 1
[9,3]
=> 10000001000 => ? => ? => ? = 10 + 1
[9,2,1]
=> 100000001010 => ? => ? => ? = 11 + 1
[9,1,1,1]
=> 1000000001110 => ? => ? => ? = 12 + 1
[8,4]
=> 1000010000 => 0001000100 => ? => ? = 9 + 1
Description
The size of the conjugacy class of a binary word.
Two words $u$ and $v$ are conjugate, if $u=w_1 w_2$ and $v=w_2 w_1$, see Section 1.3 of [1].
The following 128 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000626The minimal period of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001034The area of the parallelogram polyomino associated with the Dyck path. St000395The sum of the heights of the peaks of a Dyck path. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001554The number of distinct nonempty subtrees of a binary tree. St000385The number of vertices with out-degree 1 in a binary tree. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000026The position of the first return of a Dyck path. St000636The hull number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St000081The number of edges of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001479The number of bridges of a graph. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001342The number of vertices in the center of a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001622The number of join-irreducible elements of a lattice. St000050The depth or height of a binary tree. St000288The number of ones in a binary word. St000336The leg major index of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000384The maximal part of the shifted composition of an integer partition. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001430The number of positive entries in a signed permutation. St000505The biggest entry in the block containing the 1. St000503The maximal difference between two elements in a common block. St000019The cardinality of the support of a permutation. St000058The order of a permutation. St000863The length of the first row of the shifted shape of a permutation. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000144The pyramid weight of the Dyck path. St000189The number of elements in the poset. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000141The maximum drop size of a permutation. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000625The sum of the minimal distances to a greater element. St000703The number of deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001052The length of the exterior of a permutation. St001074The number of inversions of the cyclic embedding of a permutation. St001096The size of the overlap set of a permutation. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St000167The number of leaves of an ordered tree. St000451The length of the longest pattern of the form k 1 2. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001958The degree of the polynomial interpolating the values of a permutation. St000528The height of a poset. St000060The greater neighbor of the maximum. St001480The number of simple summands of the module J^2/J^3. St000203The number of external nodes of a binary tree. St000018The number of inversions of a permutation. St001925The minimal number of zeros in a row of an alternating sign matrix. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000259The diameter of a connected graph. St000924The number of topologically connected components of a perfect matching. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000080The rank of the poset. St001268The size of the largest ordinal summand in the poset. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000501The size of the first part in the decomposition of a permutation. St000029The depth of a permutation. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000240The number of indices that are not small excedances. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000216The absolute length of a permutation. St000653The last descent of a permutation. St000809The reduced reflection length of the permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001040The depth of the decreasing labelled binary unordered tree associated with the perfect matching. St001468The smallest fixpoint of a permutation. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset.
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