Your data matches 31 different statistics following compositions of up to 3 maps.
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Mp00230: Integer partitions parallelogram polyominoDyck paths
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.
Mp00095: Integer partitions to binary wordBinary words
Mp00135: Binary words rotate front-to-backBinary 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.
Mp00095: Integer partitions to binary wordBinary words
Mp00105: Binary words complementBinary words
Mp00135: Binary words rotate front-to-backBinary 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.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00100: Dyck paths touch compositionInteger compositions
Mp00040: Integer compositions to partitionInteger 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.
Mp00095: Integer partitions to binary wordBinary words
Mp00200: Binary words twistBinary words
Mp00097: Binary words delta morphismInteger 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.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
Mp00143: Dyck paths inverse promotionDyck 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.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00151: Permutations to cycle typeSet 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.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
Mp00149: Permutations Lehmer code rotationPermutations
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].
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00327: Dyck paths inverse Kreweras complementDyck 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.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00034: Dyck paths to binary tree: up step, left tree, down step, right treeBinary trees
Mp00010: Binary trees to ordered tree: left child = left brotherOrdered 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.