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Your data matches 31 different statistics following compositions of up to 3 maps.
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Matching statistic: St000674
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> 2
[1,1]
=> [1,1,0,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 4
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 5
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 3
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000326
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 100 => 001 => 3 = 2 + 1
[1,1]
=> 110 => 101 => 1 = 0 + 1
[3]
=> 1000 => 0001 => 4 = 3 + 1
[2,1]
=> 1010 => 0101 => 2 = 1 + 1
[1,1,1]
=> 1110 => 1101 => 1 = 0 + 1
[4]
=> 10000 => 00001 => 5 = 4 + 1
[3,1]
=> 10010 => 00101 => 3 = 2 + 1
[2,2]
=> 1100 => 1001 => 1 = 0 + 1
[2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
[1,1,1,1]
=> 11110 => 11101 => 1 = 0 + 1
[5]
=> 100000 => 000001 => 6 = 5 + 1
[4,1]
=> 100010 => 000101 => 4 = 3 + 1
[3,2]
=> 10100 => 01001 => 2 = 1 + 1
[3,1,1]
=> 100110 => 001101 => 3 = 2 + 1
[2,2,1]
=> 11010 => 10101 => 1 = 0 + 1
[2,1,1,1]
=> 101110 => 011101 => 2 = 1 + 1
[1,1,1,1,1]
=> 111110 => 111101 => 1 = 0 + 1
[6]
=> 1000000 => 0000001 => 7 = 6 + 1
[5,1]
=> 1000010 => 0000101 => 5 = 4 + 1
[4,2]
=> 100100 => 001001 => 3 = 2 + 1
[4,1,1]
=> 1000110 => 0001101 => 4 = 3 + 1
[3,3]
=> 11000 => 10001 => 1 = 0 + 1
[3,2,1]
=> 101010 => 010101 => 2 = 1 + 1
[3,1,1,1]
=> 1001110 => 0011101 => 3 = 2 + 1
[2,2,2]
=> 11100 => 11001 => 1 = 0 + 1
[2,2,1,1]
=> 110110 => 101101 => 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => 0111101 => 2 = 1 + 1
[1,1,1,1,1,1]
=> 1111110 => 1111101 => 1 = 0 + 1
[7]
=> 10000000 => 00000001 => 8 = 7 + 1
[6,1]
=> 10000010 => 00000101 => 6 = 5 + 1
[5,2]
=> 1000100 => 0001001 => 4 = 3 + 1
[5,1,1]
=> 10000110 => 00001101 => 5 = 4 + 1
[4,3]
=> 101000 => 010001 => 2 = 1 + 1
[4,2,1]
=> 1001010 => 0010101 => 3 = 2 + 1
[4,1,1,1]
=> 10001110 => 00011101 => 4 = 3 + 1
[3,3,1]
=> 110010 => 100101 => 1 = 0 + 1
[3,2,2]
=> 101100 => 011001 => 2 = 1 + 1
[3,2,1,1]
=> 1010110 => 0101101 => 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => 00111101 => 3 = 2 + 1
[2,2,2,1]
=> 111010 => 110101 => 1 = 0 + 1
[2,2,1,1,1]
=> 1101110 => 1011101 => 1 = 0 + 1
[2,1,1,1,1,1]
=> 10111110 => 01111101 => 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 11111101 => 1 = 0 + 1
[8]
=> 100000000 => 000000001 => 9 = 8 + 1
[7,1]
=> 100000010 => 000000101 => 7 = 6 + 1
[6,2]
=> 10000100 => 00001001 => 5 = 4 + 1
[6,1,1]
=> 100000110 => 000001101 => 6 = 5 + 1
[5,3]
=> 1001000 => 0010001 => 3 = 2 + 1
[5,2,1]
=> 10001010 => 00010101 => 4 = 3 + 1
[5,1,1,1]
=> 100001110 => 000011101 => 5 = 4 + 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: St000297
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 100 => 011 => 110 => 2
[1,1]
=> 110 => 001 => 010 => 0
[3]
=> 1000 => 0111 => 1110 => 3
[2,1]
=> 1010 => 0101 => 1010 => 1
[1,1,1]
=> 1110 => 0001 => 0010 => 0
[4]
=> 10000 => 01111 => 11110 => 4
[3,1]
=> 10010 => 01101 => 11010 => 2
[2,2]
=> 1100 => 0011 => 0110 => 0
[2,1,1]
=> 10110 => 01001 => 10010 => 1
[1,1,1,1]
=> 11110 => 00001 => 00010 => 0
[5]
=> 100000 => 011111 => 111110 => 5
[4,1]
=> 100010 => 011101 => 111010 => 3
[3,2]
=> 10100 => 01011 => 10110 => 1
[3,1,1]
=> 100110 => 011001 => 110010 => 2
[2,2,1]
=> 11010 => 00101 => 01010 => 0
[2,1,1,1]
=> 101110 => 010001 => 100010 => 1
[1,1,1,1,1]
=> 111110 => 000001 => 000010 => 0
[6]
=> 1000000 => 0111111 => 1111110 => 6
[5,1]
=> 1000010 => 0111101 => 1111010 => 4
[4,2]
=> 100100 => 011011 => 110110 => 2
[4,1,1]
=> 1000110 => 0111001 => 1110010 => 3
[3,3]
=> 11000 => 00111 => 01110 => 0
[3,2,1]
=> 101010 => 010101 => 101010 => 1
[3,1,1,1]
=> 1001110 => 0110001 => 1100010 => 2
[2,2,2]
=> 11100 => 00011 => 00110 => 0
[2,2,1,1]
=> 110110 => 001001 => 010010 => 0
[2,1,1,1,1]
=> 1011110 => 0100001 => 1000010 => 1
[1,1,1,1,1,1]
=> 1111110 => 0000001 => 0000010 => 0
[7]
=> 10000000 => 01111111 => 11111110 => 7
[6,1]
=> 10000010 => 01111101 => 11111010 => 5
[5,2]
=> 1000100 => 0111011 => 1110110 => 3
[5,1,1]
=> 10000110 => 01111001 => 11110010 => 4
[4,3]
=> 101000 => 010111 => 101110 => 1
[4,2,1]
=> 1001010 => 0110101 => 1101010 => 2
[4,1,1,1]
=> 10001110 => 01110001 => 11100010 => 3
[3,3,1]
=> 110010 => 001101 => 011010 => 0
[3,2,2]
=> 101100 => 010011 => 100110 => 1
[3,2,1,1]
=> 1010110 => 0101001 => 1010010 => 1
[3,1,1,1,1]
=> 10011110 => 01100001 => 11000010 => 2
[2,2,2,1]
=> 111010 => 000101 => 001010 => 0
[2,2,1,1,1]
=> 1101110 => 0010001 => 0100010 => 0
[2,1,1,1,1,1]
=> 10111110 => 01000001 => 10000010 => 1
[1,1,1,1,1,1,1]
=> 11111110 => 00000001 => 00000010 => 0
[8]
=> 100000000 => 011111111 => 111111110 => 8
[7,1]
=> 100000010 => 011111101 => 111111010 => 6
[6,2]
=> 10000100 => 01111011 => 11110110 => 4
[6,1,1]
=> 100000110 => 011111001 => 111110010 => 5
[5,3]
=> 1001000 => 0110111 => 1101110 => 2
[5,2,1]
=> 10001010 => 01110101 => 11101010 => 3
[5,1,1,1]
=> 100001110 => 011110001 => 111100010 => 4
Description
The number of leading ones in a binary word.
Matching statistic: St000475
(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
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1] => [1,1]
=> 2
[1,1]
=> [1,1,0,0]
=> [2] => [2]
=> 0
[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]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [3] => [3]
=> 0
[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]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [3] => [3]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [3,1]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [4] => [4]
=> 0
[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]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [3,1,1]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [4] => [4]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [4,1]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => [5]
=> 0
[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]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4] => [4]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [4,1]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => [5]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> 0
[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]
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1,1,1]
=> 3
[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]
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [4,1]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [4,1,1]
=> 2
[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]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => [5]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,5] => [5,1]
=> 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]
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => [5]
=> 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [6] => [6]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,6] => [6,1]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7] => [7]
=> 0
[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]
=> 6
[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]
=> 4
[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]
=> 5
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [4,1,1]
=> 2
[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]
=> 3
[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] => [4,1,1,1,1]
=> 4
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000382
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00200: Binary words —twist⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 100 => 000 => [3] => 3 = 2 + 1
[1,1]
=> 110 => 010 => [1,1,1] => 1 = 0 + 1
[3]
=> 1000 => 0000 => [4] => 4 = 3 + 1
[2,1]
=> 1010 => 0010 => [2,1,1] => 2 = 1 + 1
[1,1,1]
=> 1110 => 0110 => [1,2,1] => 1 = 0 + 1
[4]
=> 10000 => 00000 => [5] => 5 = 4 + 1
[3,1]
=> 10010 => 00010 => [3,1,1] => 3 = 2 + 1
[2,2]
=> 1100 => 0100 => [1,1,2] => 1 = 0 + 1
[2,1,1]
=> 10110 => 00110 => [2,2,1] => 2 = 1 + 1
[1,1,1,1]
=> 11110 => 01110 => [1,3,1] => 1 = 0 + 1
[5]
=> 100000 => 000000 => [6] => 6 = 5 + 1
[4,1]
=> 100010 => 000010 => [4,1,1] => 4 = 3 + 1
[3,2]
=> 10100 => 00100 => [2,1,2] => 2 = 1 + 1
[3,1,1]
=> 100110 => 000110 => [3,2,1] => 3 = 2 + 1
[2,2,1]
=> 11010 => 01010 => [1,1,1,1,1] => 1 = 0 + 1
[2,1,1,1]
=> 101110 => 001110 => [2,3,1] => 2 = 1 + 1
[1,1,1,1,1]
=> 111110 => 011110 => [1,4,1] => 1 = 0 + 1
[6]
=> 1000000 => 0000000 => [7] => 7 = 6 + 1
[5,1]
=> 1000010 => 0000010 => [5,1,1] => 5 = 4 + 1
[4,2]
=> 100100 => 000100 => [3,1,2] => 3 = 2 + 1
[4,1,1]
=> 1000110 => 0000110 => [4,2,1] => 4 = 3 + 1
[3,3]
=> 11000 => 01000 => [1,1,3] => 1 = 0 + 1
[3,2,1]
=> 101010 => 001010 => [2,1,1,1,1] => 2 = 1 + 1
[3,1,1,1]
=> 1001110 => 0001110 => [3,3,1] => 3 = 2 + 1
[2,2,2]
=> 11100 => 01100 => [1,2,2] => 1 = 0 + 1
[2,2,1,1]
=> 110110 => 010110 => [1,1,1,2,1] => 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => 0011110 => [2,4,1] => 2 = 1 + 1
[1,1,1,1,1,1]
=> 1111110 => 0111110 => [1,5,1] => 1 = 0 + 1
[7]
=> 10000000 => 00000000 => [8] => 8 = 7 + 1
[6,1]
=> 10000010 => 00000010 => [6,1,1] => 6 = 5 + 1
[5,2]
=> 1000100 => 0000100 => [4,1,2] => 4 = 3 + 1
[5,1,1]
=> 10000110 => 00000110 => [5,2,1] => 5 = 4 + 1
[4,3]
=> 101000 => 001000 => [2,1,3] => 2 = 1 + 1
[4,2,1]
=> 1001010 => 0001010 => [3,1,1,1,1] => 3 = 2 + 1
[4,1,1,1]
=> 10001110 => 00001110 => [4,3,1] => 4 = 3 + 1
[3,3,1]
=> 110010 => 010010 => [1,1,2,1,1] => 1 = 0 + 1
[3,2,2]
=> 101100 => 001100 => [2,2,2] => 2 = 1 + 1
[3,2,1,1]
=> 1010110 => 0010110 => [2,1,1,2,1] => 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => 00011110 => [3,4,1] => 3 = 2 + 1
[2,2,2,1]
=> 111010 => 011010 => [1,2,1,1,1] => 1 = 0 + 1
[2,2,1,1,1]
=> 1101110 => 0101110 => [1,1,1,3,1] => 1 = 0 + 1
[2,1,1,1,1,1]
=> 10111110 => 00111110 => [2,5,1] => 2 = 1 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 01111110 => [1,6,1] => 1 = 0 + 1
[8]
=> 100000000 => 000000000 => [9] => 9 = 8 + 1
[7,1]
=> 100000010 => 000000010 => [7,1,1] => 7 = 6 + 1
[6,2]
=> 10000100 => 00000100 => [5,1,2] => 5 = 4 + 1
[6,1,1]
=> 100000110 => 000000110 => [6,2,1] => 6 = 5 + 1
[5,3]
=> 1001000 => 0001000 => [3,1,3] => 3 = 2 + 1
[5,2,1]
=> 10001010 => 00001010 => [4,1,1,1,1] => 4 = 3 + 1
[5,1,1,1]
=> 100001110 => 000001110 => [5,3,1] => 5 = 4 + 1
Description
The first part of an integer composition.
Matching statistic: St000011
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 3 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 9 = 8 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 5 = 4 + 1
[3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000247
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 89%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 89%
Values
[2]
=> [1,0,1,0]
=> [1,2] => {{1},{2}}
=> 2
[1,1]
=> [1,1,0,0]
=> [2,1] => {{1,2}}
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => {{1},{2},{3}}
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => {{1,2,3}}
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => {{1},{2,3,4}}
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => {{1,2,3,4}}
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}}
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => {{1},{2},{3},{4},{5,6}}
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => {{1},{2},{3},{4,5,6}}
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => {{1},{2},{3,4,5,6}}
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => {{1,2,3,4}}
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => {{1,2,3,4,5}}
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => {{1},{2,3,4,5,6}}
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => {{1,2,3,4,5,6}}
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => {{1},{2},{3},{4},{5},{6},{7}}
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => {{1},{2},{3},{4},{5},{6,7}}
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => {{1},{2},{3},{4,5,6}}
=> 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => {{1},{2},{3},{4},{5,6,7}}
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,3,6,4] => {{1},{2},{3,4,5,6}}
=> 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => {{1},{2},{3},{4,5,6,7}}
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => {{1},{2,3,4,5,6}}
=> 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => {{1},{2},{3,4,5,6,7}}
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => {{1,2,3,4,5}}
=> 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => {{1,2,3,4,5,6}}
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => {{1},{2,3,4,5,6,7}}
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => {{1,2,3,4,5,6,7}}
=> 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 6
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => {{1},{2},{3},{4},{5,6,7}}
=> 4
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 5
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,5,6,3,4] => {{1},{2},{3,5},{4,6}}
=> 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => {{1},{2},{3},{4,5,6,7}}
=> 3
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 4
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => {{1,2,3,4,5}}
=> 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => {{1},{2,4},{3,5,6}}
=> 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => {{1},{2},{3,4,5,6}}
=> 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => {{1},{2},{3,4,5,6,7}}
=> 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 3
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 5
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,5,8,6] => {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 4
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,4,7,8,5] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 3
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,5,6] => {{1},{2},{3},{4},{5,7},{6,8}}
=> ? = 4
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,4,8,5] => {{1},{2},{3},{4,6},{5,7,8}}
=> ? = 3
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 4
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,2,5,6,3,7,8,4] => {{1},{2},{3,5},{4,6,7,8}}
=> ? = 2
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,4,5,8,6] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 3
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,2,6,7,8,3] => {{1},{2,4},{3,5,6,7,8}}
=> ? = 1
[7,4]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,6,7,8,4,5] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 3
[6,3,2]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,3,6,8,4,5,7] => {{1},{2},{3},{4,6},{5,7,8}}
=> ? = 3
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,2,5,7,3,4,8,6] => {{1},{2},{3,5},{4,6,7,8}}
=> ? = 2
[5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,4,5,6] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 3
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,4,6,2,3,7,8,5] => {{1},{2,4},{3,5,6,7,8}}
=> ? = 1
[7,5]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,8,3,4] => {{1},{2},{3,5,7},{4,6,8}}
=> ? = 2
[6,5,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,4,5,6,7,2,8,3] => {{1},{2,4,6},{3,5,7,8}}
=> ? = 1
[6,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,4,5,6,7] => {{1},{2},{3},{4,5,6,7,8}}
=> ? = 3
[5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,2,5,7,8,3,4,6] => {{1},{2},{3,5,6,8},{4,7}}
=> ? = 2
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ?
=> ? = 1
[4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,6,7,8,3,4,5] => {{1},{2},{3,6},{4,7},{5,8}}
=> ? = 2
[3,2,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,5,6,7,2,3,8,4] => {{1},{2,5},{3,6},{4,7,8}}
=> ? = 1
[6,5,2]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,4,5,6,8,2,3,7] => {{1},{2,4,6},{3,5,7,8}}
=> ? = 1
[6,4,3]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,2,5,8,3,4,6,7] => {{1},{2},{3,5},{4,6,7,8}}
=> ? = 2
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,4,7,2,3,5,8,6] => {{1},{2,4},{3,5,6,7,8}}
=> ? = 1
[4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,4,6,7,8,2,3,5] => {{1},{2,3,4,6,7},{5,8}}
=> ? = 1
[6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[6,4,4]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,2,7,8,3,4,5,6] => {{1},{2},{3,5,7},{4,6,8}}
=> ? = 2
[5,4,4,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,6,7,2,3,4,8,5] => {{1},{2,4,6},{3,5,7,8}}
=> ? = 1
[5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,4,7,2,8,3,5,6] => {{1},{2,4},{3,5,6,7,8}}
=> ? = 1
[4,3,3,2,2]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,6,2,7,8,3,4,5] => {{1},{2,3,6},{4,7},{5,8}}
=> ? = 1
[5,4,4,2]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,6,7,2,8,3,4,5] => {{1},{2,3,4,6,7},{5,8}}
=> ? = 1
[5,4,3,3]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,4,8,2,3,5,6,7] => {{1},{2,4},{3,5,6,7,8}}
=> ? = 1
[5,4,4,3]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,2,3,4,5,7] => {{1},{2,4,6},{3,5,7,8}}
=> ? = 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000007
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => [3,2,1] => 3 = 2 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [2,1,3] => 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [4,3,2,1] => 4 = 3 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => [2,3,1] => 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,3,4] => 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [5,4,3,2,1] => 5 = 4 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,4,2,1] => 3 = 2 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,1,4] => 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,3,4,5] => 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => [6,5,4,3,2,1] => 6 = 5 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [4,5,3,2,1] => 4 = 3 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,2,4,1] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,4,3,1] => 3 = 2 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,1,4] => 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,5,1,3,4] => 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [2,1,3,4,5,6] => 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [7,6,5,4,3,2,1] => 7 = 6 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => [5,6,4,3,2,1] => 5 = 4 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [4,3,5,2,1] => 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,5,4,2,1] => 4 = 3 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [4,3,2,1,5] => 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [2,3,4,1] => 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [2,5,4,1,3] => 3 = 2 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2,1,4,5] => 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [2,4,1,3,5] => 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => [2,6,1,3,4,5] => 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => [2,1,3,4,5,6,7] => 1 = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => [8,7,6,5,4,3,2,1] => 8 = 7 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => [6,7,5,4,3,2,1] => ? = 5 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => [5,4,6,3,2,1] => 4 = 3 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => [4,6,5,3,2,1] => 5 = 4 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [4,3,2,5,1] => 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [3,4,5,2,1] => 3 = 2 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,5,4,3,1] => 4 = 3 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,4,2,1,5] => 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,4,5,1,3] => 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => [2,6,5,1,3,4] => 3 = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,3,1,4,5] => 1 = 0 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [2,5,1,3,4,6] => 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => [2,7,1,3,4,5,6] => 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8] => 1 = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => [9,8,7,6,5,4,3,2,1] => 9 = 8 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => [7,8,6,5,4,3,2,1] => ? = 6 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => [6,5,7,4,3,2,1] => ? = 4 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => [5,7,6,4,3,2,1] => ? = 5 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => [5,4,3,6,2,1] => 3 = 2 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => [4,5,6,3,2,1] => 4 = 3 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,5,4,3,1,6] => [3,6,5,4,2,1] => 5 = 4 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => [5,4,3,2,1,6] => 1 = 0 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,4,2,5,1] => 2 = 1 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,5,4,1] => 3 = 2 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,5,3,1] => 3 = 2 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => [2,7,6,1,3,4,5] => ? = 2 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,4,3,6,2,1,7] => [6,5,4,7,3,2,1] => ? = 3 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,5,6,3,2,1,7] => [5,6,7,4,3,2,1] => ? = 4 + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,5,4,7,3,2,1,8] => [7,6,5,8,4,3,2,1] => ? = 4 + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => [6,5,4,3,7,2,1] => ? = 2 + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [4,5,3,6,2,1,7] => [5,6,4,7,3,2,1] => ? = 3 + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [4,3,6,5,2,1,7] => [5,4,7,6,3,2,1] => ? = 4 + 1
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => [6,5,4,3,2,7,1] => ? = 1 + 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [4,5,3,2,6,1,7] => [5,6,4,3,7,2,1] => ? = 2 + 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [4,3,5,6,2,1,7] => [5,4,6,7,3,2,1] => ? = 3 + 1
[3,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,6,7,5,4,3] => [3,2,7,1,4,5,6] => ? = 1 + 1
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,6,7,5,4,2] => [2,4,7,1,3,5,6] => ? = 1 + 1
[7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [6,5,4,3,2,7,1,8] => [7,6,5,4,3,8,2,1] => ? = 2 + 1
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [4,5,3,2,1,6,7] => [5,6,4,3,2,7,1] => ? = 1 + 1
[6,4,2]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [4,3,5,2,6,1,7] => [5,4,6,3,7,2,1] => ? = 2 + 1
[6,3,3]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> [4,3,2,6,5,1,7] => [5,4,3,7,6,2,1] => ? = 3 + 1
[4,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,5,7,4,3] => [3,2,7,6,1,4,5] => ? = 2 + 1
[3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [2,1,5,7,6,4,3] => [3,2,6,1,4,5,7] => ? = 0 + 1
[3,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [2,3,7,1,4,5,6] => ? = 1 + 1
[6,5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [4,3,5,2,1,6,7] => [5,4,6,3,2,7,1] => ? = 1 + 1
[6,4,3]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [4,3,2,5,6,1,7] => [5,4,3,6,7,2,1] => ? = 2 + 1
[5,4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [4,3,2,1,6,7,5] => [5,4,3,2,7,1,6] => ? = 1 + 1
[4,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [3,4,2,1,7,6,5] => [4,5,3,2,1,6,7] => ? = 0 + 1
[4,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,2,1,6,7,5,4] => [4,3,2,7,1,5,6] => ? = 1 + 1
[4,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [2,1,5,6,7,4,3] => [3,2,6,7,1,4,5] => ? = 1 + 1
[3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [2,3,1,7,6,5,4] => [3,4,2,1,5,6,7] => ? = 0 + 1
[3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [2,1,4,7,6,5,3] => [3,2,5,1,4,6,7] => ? = 0 + 1
[3,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,1,7,8,6,5,4,3] => [3,2,8,1,4,5,6,7] => ? = 1 + 1
[6,5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,3,2,5,1,6,7] => [5,4,3,6,2,7,1] => ? = 1 + 1
[6,4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,7] => [5,4,3,2,7,6,1] => ? = 2 + 1
[5,4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,4,2,1,6,7,5] => [4,5,3,2,7,1,6] => ? = 1 + 1
[5,3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [3,2,1,6,5,7,4] => [4,3,2,7,6,1,5] => ? = 2 + 1
[4,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [3,2,4,1,7,6,5] => [4,3,5,2,1,6,7] => ? = 0 + 1
[4,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [3,2,1,5,7,6,4] => [4,3,2,6,1,5,7] => ? = 0 + 1
[4,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1,6,7,5,4] => [3,4,2,7,1,5,6] => ? = 1 + 1
[4,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,1,4,6,7,5,3] => [3,2,5,7,1,4,6] => ? = 1 + 1
[3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,6,5,4] => [3,2,4,1,5,6,7] => ? = 0 + 1
[6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,3,2,1,5,6,7] => [5,4,3,2,6,7,1] => ? = 1 + 1
[5,4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,2,4,1,6,7,5] => [4,3,5,2,7,1,6] => ? = 1 + 1
[5,4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [3,2,1,5,6,7,4] => [4,3,2,6,7,1,5] => ? = 1 + 1
[4,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,7,6,5] => [4,3,2,5,1,6,7] => ? = 0 + 1
[4,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,6,7,5,4] => [3,2,4,7,1,5,6] => ? = 1 + 1
[6,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [5,4,3,2,1,7,8,6] => [6,5,4,3,2,8,1,7] => ? = 1 + 1
[5,4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [3,2,1,4,6,7,5] => [4,3,2,5,7,1,6] => ? = 1 + 1
[4,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,1,7,8,6,5,4] => [4,3,2,8,1,5,6,7] => ? = 1 + 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St001107
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 89%
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 89%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 6
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 4
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 5
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> 3
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 4
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 3
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 5
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 4
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? = 3
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> ? = 4
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 3
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 4
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> ? = 2
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 3
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> ? = 2
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
[7,4]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 3
[6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> ? = 2
[6,3,2]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> ? = 3
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 1
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> ? = 2
[5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 3
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 2
[3,2,2,2,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 1
[7,5]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 2
[6,5,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 1
[6,4,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 2
[6,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 1
[5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 2
[5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> ? = 2
[4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
[4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? = 2
[3,2,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 1
[7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[6,5,2]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 1
[6,4,3]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 2
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1
[5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> ? = 1
[5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 2
[4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 1
[4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> ? = 1
[3,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
[6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 1
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000974
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000974: Ordered trees ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 89%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000974: Ordered trees ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 89%
Values
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> [[[]]]
=> 2
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> [[],[]]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [[[[]]]]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [[[],[]]]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[.,[.,.]],.]
=> [[[]],[]]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [[[.,.],.],.]
=> [[],[],[]]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,.]]]],.]
=> [[[[[]]]],[]]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [[[[[[[]]]]]]]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [[[[[[],[]]]]]]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> [[[[[[]],[]]]]]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> [[[[[[]]],[]]]]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[[.,.],.],.],.]
=> [[],[],[],[]]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[.,.],[.,[.,.]]],.]
=> [[],[[[]]],[]]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> [[[[[[]]]],[]]]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> [[[[[[]]]]],[]]
=> 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [[[[[[[[]]]]]]]]
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [[[[[[[],[]]]]]]]
=> 5
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> [[[[[],[],[]]]]]
=> 3
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [[[[[[[]],[]]]]]]
=> 4
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> [[[[],[[]],[]]]]
=> 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [[[[[[[]]],[]]]]]
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[.,[.,.]],[.,.]],.]
=> [[[]],[[]],[]]
=> 0
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[[.,.],[.,[.,.]]],.]]
=> [[[],[[[]]],[]]]
=> 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [[[[[[[]]]],[]]]]
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[.,.],.],[.,.]],.]
=> [[],[],[[]],[]]
=> 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[.,.],[.,[.,[.,.]]]],.]
=> [[],[[[[]]]],[]]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [[[[[[[]]]]],[]]]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> [[[[[[[]]]]]],[]]
=> 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [[[[[[[[[]]]]]]]]]
=> ? = 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [[[[[[[[],[]]]]]]]]
=> ? = 6
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [[[[[[],[],[]]]]]]
=> 4
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [[[[[[[[]],[]]]]]]]
=> ? = 5
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> [[[[[]],[],[]]]]
=> 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [[[[[],[[]],[]]]]]
=> 3
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [[[[[[[[]]],[]]]]]]
=> ? = 4
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[.,[.,[.,.]]],.],.]
=> [[[[]]],[],[]]
=> 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> [[[[]],[[]],[]]]
=> 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> [[[[],[],[],[]]]]
=> 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [[[[],[[[]]],[]]]]
=> 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [[[[[[[[]]]],[]]]]]
=> ? = 3
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [[[[[[[[]]]]],[]]]]
=> ? = 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [[[[[[[[]]]]]],[]]]
=> ? = 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [[[[[[[],[],[]]]]]]]
=> ? = 5
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [[[[[[],[[]],[]]]]]]
=> ? = 4
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [[[[[],[[[]]],[]]]]]
=> ? = 3
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [[[[],[[[[]]]],[]]]]
=> ? = 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [[[],[[[[[]]]]],[]]]
=> ? = 1
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [[[[[[[]],[],[]]]]]]
=> ? = 4
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [[[[[[]],[[]],[]]]]]
=> ? = 3
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [[[[[[],[],[],[]]]]]]
=> ? = 4
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [[[[[]],[[[]]],[]]]]
=> ? = 2
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [[[[[],[],[[]],[]]]]]
=> ? = 3
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [[[[]],[[[[]]]],[]]]
=> ? = 1
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [[[[],[],[[[]]],[]]]]
=> ? = 2
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [[[],[],[[[[]]]],[]]]
=> ? = 1
[7,4]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[.,[[[.,[.,[.,.]]],.],.]]]]
=> [[[[[[[]]],[],[]]]]]
=> ? = 3
[6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [.,[.,[[[.,[.,[.,.]]],[.,.]],.]]]
=> [[[[[[]]],[[]],[]]]]
=> ? = 2
[6,3,2]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [.,[.,[.,[[[.,[[.,.],.]],.],.]]]]
=> [[[[[[],[]],[],[]]]]]
=> ? = 3
[5,4,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [[[[[]]],[[[]]],[]]]
=> ? = 1
[5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [.,[.,[[[.,[[.,.],.]],[.,.]],.]]]
=> [[[[[],[]],[[]],[]]]]
=> ? = 2
[5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [.,[.,[.,[[[[.,[.,.]],.],.],.]]]]
=> [[[[[[]],[],[],[]]]]]
=> ? = 3
[4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [.,[[[.,[[.,.],.]],[.,[.,.]]],.]]
=> [[[[],[]],[[[]]],[]]]
=> ? = 1
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [.,[.,[[[[.,[.,.]],.],[.,.]],.]]]
=> [[[[[]],[],[[]],[]]]]
=> ? = 2
[3,2,2,2,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [.,[[[[.,[.,.]],.],[.,[.,.]]],.]]
=> [[[[]],[],[[[]]],[]]]
=> ? = 1
[7,5]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [.,[.,[[[.,[.,[.,[.,.]]]],.],.]]]
=> [[[[[[[]]]],[],[]]]]
=> ? = 2
[6,5,1]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [[[[[[]]]],[[]],[]]]
=> ? = 1
[6,4,2]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [.,[.,[[[.,[.,[[.,.],.]]],.],.]]]
=> [[[[[[],[]]],[],[]]]]
=> ? = 2
[6,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [[[[[],[],[],[],[]]]]]
=> ? = 3
[5,4,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> [[[[[],[]]],[[]],[]]]
=> ? = 1
[5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [.,[.,[[[[[.,.],.],.],[.,.]],.]]]
=> [[[[],[],[],[[]],[]]]]
=> ? = 2
[5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [.,[.,[[[.,[[.,[.,.]],.]],.],.]]]
=> [[[[[[]],[]],[],[]]]]
=> ? = 2
[4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],[.,[.,.]]],.]]
=> [[[],[],[],[[[]]],[]]]
=> ? = 1
[4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [[[[[]],[]],[[]],[]]]
=> ? = 1
[4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [.,[.,[[[[.,[.,[.,.]]],.],.],.]]]
=> [[[[[[]]],[],[],[]]]]
=> ? = 2
[3,2,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [.,[[[[.,[.,[.,.]]],.],[.,.]],.]]
=> [[[[[]]],[],[[]],[]]]
=> ? = 1
[7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [.,[[[.,[.,[.,[.,[.,.]]]]],.],.]]
=> [[[[[[[]]]]],[],[]]]
=> ? = 1
[6,5,2]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [.,[[[.,[.,[.,[[.,.],.]]]],.],.]]
=> [[[[[[],[]]]],[],[]]]
=> ? = 1
[6,4,3]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [.,[.,[[[.,[[[.,.],.],.]],.],.]]]
=> [[[[[],[],[]],[],[]]]]
=> ? = 2
[5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [.,[[[.,[[[.,.],.],.]],[.,.]],.]]
=> [[[[],[],[]],[[]],[]]]
=> ? = 1
[5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [.,[[[.,[.,[[.,[.,.]],.]]],.],.]]
=> [[[[[[]],[]]],[],[]]]
=> ? = 1
[5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [.,[.,[[[[[.,.],[.,.]],.],.],.]]]
=> [[[[],[[]],[],[],[]]]]
=> ? = 2
[4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [.,[[[[[.,.],[.,.]],.],[.,.]],.]]
=> [[[],[[]],[],[[]],[]]]
=> ? = 1
[4,3,2,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [.,[[[.,[[.,[.,[.,.]]],.]],.],.]]
=> [[[[[[]]],[]],[],[]]]
=> ? = 1
[3,2,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [[[[[[]]]],[],[],[]]]
=> ? = 1
[6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [.,[[[.,[.,[[[.,.],.],.]]],.],.]]
=> [[[[[],[],[]]],[],[]]]
=> ? = 1
Description
The length of the trunk of an ordered tree.
This is the length of the path from the root to the first vertex which has not exactly one child.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000022The number of fixed points of a permutation. St000873The aix statistic of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000993The multiplicity of the largest part of an integer partition. St000237The number of small exceedances. St000461The rix statistic of a permutation. St000260The radius of a connected graph. St000117The number of centered tunnels of a Dyck path. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000164The number of short pairs. St000315The number of isolated vertices of a graph. St000338The number of pixed points of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001691The number of kings in a graph. St000894The trace of an alternating sign matrix. St000239The number of small weak excedances. St000335The difference of lower and upper interactions. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra.
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