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Your data matches 64 different statistics following compositions of up to 3 maps.
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Matching statistic: St000383
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1] => 1
[1,1,0,0]
=> [2] => 2
[1,0,1,0,1,0]
=> [1,1,1] => 1
[1,0,1,1,0,0]
=> [1,2] => 2
[1,1,0,0,1,0]
=> [2,1] => 1
[1,1,0,1,0,0]
=> [3] => 3
[1,1,1,0,0,0]
=> [3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 3
[1,0,1,1,1,0,0,0]
=> [1,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1
[1,1,0,1,0,1,0,0]
=> [4] => 4
[1,1,0,1,1,0,0,0]
=> [4] => 4
[1,1,1,0,0,0,1,0]
=> [3,1] => 1
[1,1,1,0,0,1,0,0]
=> [4] => 4
[1,1,1,0,1,0,0,0]
=> [4] => 4
[1,1,1,1,0,0,0,0]
=> [4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 5
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 5
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 5
Description
The last part of an integer composition.
Matching statistic: St000382
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,1] => [1,1] => 1
[1,1,0,0]
=> [2] => [2] => 2
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => 2
[1,1,0,0,1,0]
=> [2,1] => [1,2] => 1
[1,1,0,1,0,0]
=> [3] => [3] => 3
[1,1,1,0,0,0]
=> [3] => [3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,2] => 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1] => 3
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2,1] => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,3] => 1
[1,1,0,1,0,1,0,0]
=> [4] => [4] => 4
[1,1,0,1,1,0,0,0]
=> [4] => [4] => 4
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 1
[1,1,1,0,0,1,0,0]
=> [4] => [4] => 4
[1,1,1,0,1,0,0,0]
=> [4] => [4] => 4
[1,1,1,1,0,0,0,0]
=> [4] => [4] => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,1,1,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,1,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,3] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1] => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,3] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,2,1,1] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,3,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [2,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5] => 5
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5] => 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5] => 5
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5] => 5
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5] => 5
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,2,1,1,1,1] => [1,1,1,2,2,1,1,1] => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,5,1,1,1,1] => [1,1,5,1,1,1] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,1,1,1,2,1,1] => [1,2,1,1,1,1,2,1] => ? = 1
[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,6,1] => [1,2,1,6] => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1,1] => [1,2,2,1,1,1,1,1] => ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,5,1,1,1,1] => [1,1,5,1,1,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,5,1,1,1,1] => [1,1,5,1,1,1] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,1,6,1] => [1,2,1,6] => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,6,2,1] => [1,1,6,2] => ? = 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> [1,6,2,1] => [1,1,6,2] => ? = 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,1,0,0,1,0]
=> [1,6,2,1] => [1,1,6,2] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,5,3,1] => [1,1,5,3] => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
=> [1,6,2,1] => [1,1,6,2] => ? = 1
Description
The first part of an integer composition.
Matching statistic: St000297
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1 => 1
[1,0,1,0]
=> [1,1] => [2] => 10 => 1
[1,1,0,0]
=> [2] => [1,1] => 11 => 2
[1,0,1,0,1,0]
=> [1,1,1] => [3] => 100 => 1
[1,0,1,1,0,0]
=> [1,2] => [1,2] => 110 => 2
[1,1,0,0,1,0]
=> [2,1] => [2,1] => 101 => 1
[1,1,0,1,0,0]
=> [3] => [1,1,1] => 111 => 3
[1,1,1,0,0,0]
=> [3] => [1,1,1] => 111 => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 1000 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => 1100 => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => 1010 => 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [1,1,2] => 1110 => 3
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => 1110 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => 1001 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => 1101 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [2,1,1] => 1011 => 1
[1,1,0,1,0,1,0,0]
=> [4] => [1,1,1,1] => 1111 => 4
[1,1,0,1,1,0,0,0]
=> [4] => [1,1,1,1] => 1111 => 4
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => 1011 => 1
[1,1,1,0,0,1,0,0]
=> [4] => [1,1,1,1] => 1111 => 4
[1,1,1,0,1,0,0,0]
=> [4] => [1,1,1,1] => 1111 => 4
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1111 => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => 10000 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => 11000 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => 10100 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => 11100 => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => 11100 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => 10010 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => 11010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [2,1,2] => 10110 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1,1,2] => 11110 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1,1,2] => 11110 => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => 10110 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1,1,2] => 11110 => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1,1,2] => 11110 => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => 11110 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => 10001 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => 11001 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => 10101 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1,2,1] => 11101 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => 11101 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [3,1,1] => 10011 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,2,1,1] => 11011 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [2,1,1,1] => 10111 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1,1,1,1,1] => 11111 => 5
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1,1,1,1,1] => 11111 => 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [2,1,1,1] => 10111 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1,1,1,1,1] => 11111 => 5
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1,1,1,1,1] => 11111 => 5
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1,1,1,1,1] => 11111 => 5
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,2,2] => [1,2,7] => 1101000000 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,2,1,1,1,1] => [5,2,3] => 1000010100 => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,4,4] => [1,1,1,2,1,1,3] => 1111011100 => ? = 4
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,7,1] => [2,1,1,1,1,1,3] => 1011111100 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,1,1,2] => [1,8,1] => 1100000001 => ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,1,1,1,2,1,1] => [3,6,1] => 1001000001 => ? = 1
[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,6,1] => [2,1,1,1,1,3,1] => 1011111001 => ? = 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,2,1,1,1,1,1,1] => [7,2,1] => 1000000101 => ? = 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,2,3,3] => [1,1,2,1,2,2,1] => 1110110101 => ? = 3
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,2,3] => [1,1,2,2,2,1,1] => 1110101011 => ? = 3
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,3,2,2] => [1,2,2,1,2,1,1] => 1101011011 => ? = 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,1,4] => [1,1,1,4,1,1,1] => 1111000111 => ? = 4
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,4,1,1] => [3,1,1,2,1,1,1] => 1001110111 => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => 1111101111 => ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [8,1,1] => [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => 1111101111 => ? = 5
[1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => 1111101111 => ? = 5
[1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => 1111101111 => ? = 5
[1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [5,5] => [1,1,1,1,2,1,1,1,1] => 1111101111 => ? = 5
[1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [9,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
Description
The number of leading ones in a binary word.
Matching statistic: St000439
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [2] => [2] => [1,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0]
=> [3] => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,0,1,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,1,0,0,0,0]
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,3,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,3,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,3,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,3,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,1,4] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,2,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,4,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,2,1,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,2,1,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [2,3,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [3,1,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,1,0,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [3,2,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [3,1,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,1,0,0]
=> [3,2,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3 + 1
[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] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,4,1] => [1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,6,1] => [1,1,1,6] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,5,1] => [1,1,2,5] => [1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,4,1,1,1,1] => [1,1,4,1,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,4,1,3] => [3,1,4,1] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,6,2] => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,5,1] => [1,2,1,5] => [1,0,1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,1] => [1,3,1,4] => [1,0,1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000326
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 90%
Values
[1,0]
=> [1] => 1 => 1 => 1
[1,0,1,0]
=> [1,1] => 11 => 11 => 1
[1,1,0,0]
=> [2] => 10 => 01 => 2
[1,0,1,0,1,0]
=> [1,1,1] => 111 => 111 => 1
[1,0,1,1,0,0]
=> [1,2] => 110 => 011 => 2
[1,1,0,0,1,0]
=> [2,1] => 101 => 101 => 1
[1,1,0,1,0,0]
=> [3] => 100 => 001 => 3
[1,1,1,0,0,0]
=> [3] => 100 => 001 => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 1111 => 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0111 => 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1011 => 1
[1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 0011 => 3
[1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0011 => 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1101 => 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 0101 => 2
[1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,0,1,0,1,0,0]
=> [4] => 1000 => 0001 => 4
[1,1,0,1,1,0,0,0]
=> [4] => 1000 => 0001 => 4
[1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1001 => 1
[1,1,1,0,0,1,0,0]
=> [4] => 1000 => 0001 => 4
[1,1,1,0,1,0,0,0]
=> [4] => 1000 => 0001 => 4
[1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0001 => 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 11111 => 11111 => 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 11110 => 01111 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 11101 => 10111 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 11100 => 00111 => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 11100 => 00111 => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 11011 => 11011 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 11010 => 01011 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 11001 => 10011 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 00011 => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 10111 => 11101 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 10110 => 01101 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 10101 => 10101 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 00101 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 00101 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 10011 => 11001 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 01001 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 10001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 00001 => 5
[1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 00001 => 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 10001 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 00001 => 5
[1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 00001 => 5
[1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 00001 => 5
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 10
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 10
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 10
[1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 10
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => 10000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 10
[1,0,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 10
[1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 10
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 10
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,1,1,1,0,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 10
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => 1000000001 => ? = 1
[1,0,1,1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 10
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,1] => 1000000011 => 1100000001 => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,8,1] => 1100000001 => 1000000011 => ? = 1
[1,0,1,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => 00000000011 => ? = 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 => 1111111111 => ? = 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 => 00000000011 => ? = 10
[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,5,1] => 1111100001 => 1000011111 => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,2,1,1,1,1] => 1110101111 => 1111010111 => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2,2,2] => 1110101010 => 0101010111 => ? = 2
[1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,7,1] => 1110000001 => 1000000111 => ? = 1
[1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,5,1,1,1,1] => 1100001111 => 1111000011 => ? = 1
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,7,1,1] => 1100000011 => 1100000011 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,1,1,1,2,1,1] => 1011111011 => 1101111101 => ? = 1
[1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,1,6,1] => 1011000001 => 1000001101 => ? = 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000678
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 70%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1
[1,0,1,0]
=> [1,1] => [2] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [2] => [1,1] => [1,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0]
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,1,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,0,1,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,0,1,1,0,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,1,0,1,0,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,0,1,1,1,0,0,0,0]
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,2,1] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,2,1,1] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,2,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,3,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,2,1,1,1] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,2,1,2] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,2,2,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,3,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,3,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,3,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1,1] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,2,1,1,2] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1,2,1] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,2,2,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,2,3,1] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,3,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,3,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,3,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,3,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,3,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,3,2,1] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,3,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,3,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,5,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,8] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1] => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8] => [2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000645
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,4,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,4,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [8,4,5,3,6,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,5,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,4,2,3,1] => ?
=> ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,7,2,3,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,2,4,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,3,2,6,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,3,2,6,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,2,7,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [6,7,4,3,5,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [5,4,6,3,7,2,8,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 7 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [8,7,3,4,5,2,6,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [7,8,3,4,5,2,6,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,6,2,7,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,5,6,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [5,4,3,6,7,2,8,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 7 - 1
[1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [4,5,3,6,7,2,8,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 7 - 1
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,7,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [4,3,5,6,7,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,2,3,4,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,2,3,5,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,2,3,5,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,2,3,6,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,2,3,6,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,2,3,6,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,2,3,7,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,2,3,7,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,2,3,7,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,2,3,8,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 7 - 1
[1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [7,6,8,3,2,4,5,1] => ?
=> ?
=> ? = 3 - 1
[1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [7,8,5,3,2,4,6,1] => ?
=> ?
=> ? = 2 - 1
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St000147
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 4 = 5 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 7 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 7 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 7 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 7 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 7 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,4,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,4,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [8,4,5,3,6,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,5,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,4,2,3,1] => ?
=> ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,7,2,3,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,2,4,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,3,2,6,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,3,2,6,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [5,6,4,7,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,2,7,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [5,4,6,7,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [6,7,5,3,4,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [5,6,7,3,4,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [6,7,4,3,5,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [5,6,4,3,7,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [8,7,3,4,5,2,6,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [7,8,3,4,5,2,6,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [6,7,3,4,5,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [8,5,3,4,6,2,7,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [7,4,3,5,6,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [5,3,4,6,7,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [4,3,5,6,7,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,2,3,4,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,2,3,5,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [8,6,7,4,2,3,5,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,2,3,6,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,2,3,6,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,2,3,6,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,2,3,7,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,2,3,7,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,2,3,7,1] => ?
=> ?
=> ? = 1 - 1
Description
The largest part of an integer partition.
Matching statistic: St000384
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 4 = 5 - 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1]
=> ? = 7 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? = 7 - 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1]
=> ? = 7 - 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? = 7 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2]
=> ? = 7 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1]
=> ? = 7 - 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,1,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 7 - 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,1,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? = 7 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? = 7 - 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,1,6,7] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1]
=> ? = 7 - 1
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? = 7 - 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,1,6,7] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2]
=> ? = 7 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1]
=> ? = 7 - 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,3,5,1,2,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 7 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? = 7 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 7 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,4,2,1,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? = 7 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,2,3,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 7 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,4,1,5,6,7] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2]
=> ? = 7 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 7 - 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 7 - 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? = 7 - 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 7 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 7 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 7 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,2,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 7 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 7 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 7 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,4,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,4,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [8,4,5,3,6,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,5,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,4,2,3,1] => ?
=> ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,7,2,3,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,3,2,4,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [5,6,7,4,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,3,2,6,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,3,2,6,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [7,5,4,6,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
[1,0,1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [6,5,4,7,3,2,8,1] => ?
=> ?
=> ? = 7 - 1
Description
The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also [[St000380]].
Matching statistic: St000784
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000784: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3 = 4 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4 = 5 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4 = 5 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 4 = 5 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 4 = 5 - 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? = 6 - 1
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? = 6 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? = 6 - 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1]
=> ? = 6 - 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2]
=> ? = 6 - 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> ? = 6 - 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,1,7] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1]
=> ? = 7 - 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? = 7 - 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,1,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [6,4,3,1]
=> ? = 7 - 1
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? = 7 - 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,1,7] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2]
=> ? = 7 - 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1]
=> ? = 7 - 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,1,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 7 - 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,1,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? = 7 - 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? = 7 - 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,1,6,7] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [6,5,3,1]
=> ? = 7 - 1
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> ? = 7 - 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,1,6,7] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2]
=> ? = 7 - 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1]
=> ? = 7 - 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [4,3,5,1,2,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? = 7 - 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [6,5,2,1]
=> ? = 7 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [4,3,2,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 7 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,4,2,1,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? = 7 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [4,2,3,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 7 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [3,2,4,1,5,6,7] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2]
=> ? = 7 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? = 7 - 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [4,3,1,2,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 7 - 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? = 7 - 1
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 7 - 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [3,2,1,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 7 - 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 7 - 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [4,1,2,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> ? = 7 - 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,2,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 7 - 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 7 - 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 7 - 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,3,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,3,4,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [8,5,6,4,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [8,6,4,5,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,3,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> [8,6,5,3,4,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [8,5,6,3,4,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [8,4,5,3,6,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [8,6,3,4,5,7,2,1] => ?
=> ?
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [6,5,7,8,4,2,3,1] => ?
=> ?
=> ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [7,8,5,4,6,2,3,1] => ?
=> ?
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [8,5,4,6,7,2,3,1] => ?
=> ?
=> ? = 1 - 1
Description
The maximum of the length and the largest part of the integer partition.
This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1].
See also [[St001214]].
The following 54 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000054The first entry of the permutation. St000019The cardinality of the support of a permutation. St000141The maximum drop size of a permutation. St000010The length of the partition. St001176The size of a partition minus its first part. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000505The biggest entry in the block containing the 1. St000026The position of the first return of a Dyck path. St000025The number of initial rises of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St000363The number of minimal vertex covers of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000653The last descent of a permutation. St001316The domatic number of a graph. St000501The size of the first part in the decomposition of a permutation. St000209Maximum difference of elements in cycles. St000844The size of the largest block in the direct sum decomposition of a permutation. St000795The mad of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000740The last entry of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000287The number of connected components of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000051The size of the left subtree of a binary tree. St000286The number of connected components of the complement of a graph. St000553The number of blocks of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000030The sum of the descent differences of a permutations. St000090The variation of a composition. St000133The "bounce" of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000316The number of non-left-to-right-maxima of a permutation. 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. St001480The number of simple summands of the module J^2/J^3. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001645The pebbling number of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000840The number of closers smaller than the largest opener in a perfect matching. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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