Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000977
(load all 2 compositions to match this statistic)
Mapping
St000977: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => 0
[1,1,0,0]
 => 4
[1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => 8
[1,1,0,0,1,0]
 => 4
[1,1,0,1,0,0]
 => 6
[1,1,1,0,0,0]
 => 12
[1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 12
[1,0,1,1,0,0,1,0]
 => 8
[1,0,1,1,0,1,0,0]
 => 10
[1,0,1,1,1,0,0,0]
 => 20
[1,1,0,0,1,0,1,0]
 => 4
[1,1,0,0,1,1,0,0]
 => 16
[1,1,0,1,0,0,1,0]
 => 6
[1,1,0,1,0,1,0,0]
 => 8
[1,1,0,1,1,0,0,0]
 => 18
[1,1,1,0,0,0,1,0]
 => 12
[1,1,1,0,0,1,0,0]
 => 14
[1,1,1,0,1,0,0,0]
 => 16
[1,1,1,1,0,0,0,0]
 => 24
[1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => 16
[1,0,1,0,1,1,0,0,1,0]
 => 12
[1,0,1,0,1,1,0,1,0,0]
 => 14
[1,0,1,0,1,1,1,0,0,0]
 => 28
[1,0,1,1,0,0,1,0,1,0]
 => 8
[1,0,1,1,0,0,1,1,0,0]
 => 24
[1,0,1,1,0,1,0,0,1,0]
 => 10
[1,0,1,1,0,1,0,1,0,0]
 => 12
[1,0,1,1,0,1,1,0,0,0]
 => 26
[1,0,1,1,1,0,0,0,1,0]
 => 20
[1,0,1,1,1,0,0,1,0,0]
 => 22
[1,0,1,1,1,0,1,0,0,0]
 => 24
[1,0,1,1,1,1,0,0,0,0]
 => 36
[1,1,0,0,1,0,1,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => 20
[1,1,0,0,1,1,0,0,1,0]
 => 16
[1,1,0,0,1,1,0,1,0,0]
 => 18
[1,1,0,0,1,1,1,0,0,0]
 => 32
[1,1,0,1,0,0,1,0,1,0]
 => 6
[1,1,0,1,0,0,1,1,0,0]
 => 22
[1,1,0,1,0,1,0,0,1,0]
 => 8
[1,1,0,1,0,1,0,1,0,0]
 => 10
[1,1,0,1,0,1,1,0,0,0]
 => 24
[1,1,0,1,1,0,0,0,1,0]
 => 18
[1,1,0,1,1,0,0,1,0,0]
 => 20
[1,1,0,1,1,0,1,0,0,0]
 => 22
[1,1,0,1,1,1,0,0,0,0]
 => 34
[1,1,1,0,0,0,1,0,1,0]
 => 12
Description
MacMahon's equal index of a Dyck path. This is the sum of the positions of double (up- or down-)steps of a Dyck path, see [1, p. 135].
Matching statistic: St000391
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 25%
Values
[1,0,1,0]
 => 1010 => 0000 => 0000 => 0
[1,1,0,0]
 => 1100 => 0101 => 1010 => 4
[1,0,1,0,1,0]
 => 101010 => 000000 => 000000 => 0
[1,0,1,1,0,0]
 => 101100 => 010100 => 001010 => 8
[1,1,0,0,1,0]
 => 110010 => 000101 => 101000 => 4
[1,1,0,1,0,0]
 => 110100 => 010001 => 100010 => 6
[1,1,1,0,0,0]
 => 111000 => 011011 => 110110 => 12
[1,0,1,0,1,0,1,0]
 => 10101010 => 00000000 => 00000000 => 0
[1,0,1,0,1,1,0,0]
 => 10101100 => 01010000 => 00001010 => 12
[1,0,1,1,0,0,1,0]
 => 10110010 => 00010100 => 00101000 => 8
[1,0,1,1,0,1,0,0]
 => 10110100 => 01000100 => 00100010 => 10
[1,0,1,1,1,0,0,0]
 => 10111000 => 01101100 => 00110110 => 20
[1,1,0,0,1,0,1,0]
 => 11001010 => 00000101 => 10100000 => 4
[1,1,0,0,1,1,0,0]
 => 11001100 => 01010101 => 10101010 => 16
[1,1,0,1,0,0,1,0]
 => 11010010 => 00010001 => 10001000 => 6
[1,1,0,1,0,1,0,0]
 => 11010100 => 01000001 => 10000010 => 8
[1,1,0,1,1,0,0,0]
 => 11011000 => 01101001 => 10010110 => 18
[1,1,1,0,0,0,1,0]
 => 11100010 => 00011011 => 11011000 => 12
[1,1,1,0,0,1,0,0]
 => 11100100 => 01001011 => 11010010 => 14
[1,1,1,0,1,0,0,0]
 => 11101000 => 01100011 => 11000110 => 16
[1,1,1,1,0,0,0,0]
 => 11110000 => 01110111 => 11101110 => 24
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0000000000 => 0000000000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0101000000 => 0000001010 => 16
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0001010000 => 0000101000 => 12
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0100010000 => 0000100010 => 14
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0110110000 => 0000110110 => 28
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0000010100 => 0010100000 => 8
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0101010100 => 0010101010 => 24
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0001000100 => 0010001000 => 10
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0100000100 => 0010000010 => 12
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0110100100 => 0010010110 => 26
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0001101100 => 0011011000 => ? = 20
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0100101100 => 0011010010 => ? = 22
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0110001100 => 0011000110 => ? = 24
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0111011100 => 0011101110 => ? = 36
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 0000000101 => 1010000000 => ? = 4
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 0101000101 => 1010001010 => ? = 20
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 0001010101 => 1010101000 => ? = 16
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 0100010101 => 1010100010 => ? = 18
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 0110110101 => 1010110110 => ? = 32
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 0000010001 => 1000100000 => ? = 6
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 0101010001 => 1000101010 => ? = 22
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 0001000001 => 1000001000 => ? = 8
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 0100000001 => 1000000010 => ? = 10
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 0110100001 => 1000010110 => ? = 24
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 0001101001 => 1001011000 => ? = 18
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 0100101001 => 1001010010 => ? = 20
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 0110001001 => 1001000110 => ? = 22
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0111011001 => 1001101110 => ? = 34
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 0000011011 => 1101100000 => ? = 12
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 0101011011 => 1101101010 => ? = 28
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 0001001011 => 1101001000 => ? = 14
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 0100001011 => 1101000010 => ? = 16
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0110101011 => 1101010110 => ? = 30
[1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 0001100011 => 1100011000 => ? = 16
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 0100100011 => 1100010010 => ? = 18
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 0110000011 => 1100000110 => ? = 20
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 0111010011 => 1100101110 => ? = 32
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 0001110111 => 1110111000 => ? = 24
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0100110111 => 1110110010 => ? = 26
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 0110010111 => 1110100110 => ? = 28
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 0111000111 => 1110001110 => ? = 30
[1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 0111101111 => 1111011110 => ? = 40
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => 000000000000 => 000000000000 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => 010100000000 => 000000001010 => ? = 20
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => 000101000000 => 000000101000 => ? = 16
[1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => 010001000000 => 000000100010 => ? = 18
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => 011011000000 => 000000110110 => ? = 36
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => 000001010000 => 000010100000 => ? = 12
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => 010101010000 => 000010101010 => ? = 32
[1,0,1,0,1,1,0,1,0,0,1,0]
 => 101011010010 => 000100010000 => 000010001000 => ? = 14
[1,0,1,0,1,1,0,1,0,1,0,0]
 => 101011010100 => 010000010000 => 000010000010 => ? = 16
[1,0,1,0,1,1,0,1,1,0,0,0]
 => 101011011000 => 011010010000 => 000010010110 => ? = 34
[1,0,1,0,1,1,1,0,0,0,1,0]
 => 101011100010 => 000110110000 => 000011011000 => ? = 28
[1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => 010010110000 => 000011010010 => ? = 30
[1,0,1,0,1,1,1,0,1,0,0,0]
 => 101011101000 => 011000110000 => 000011000110 => ? = 32
[1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => 011101110000 => 000011101110 => ? = 48
[1,0,1,1,0,0,1,0,1,0,1,0]
 => 101100101010 => 000000010100 => 001010000000 => ? = 8
[1,0,1,1,0,0,1,0,1,1,0,0]
 => 101100101100 => 010100010100 => 001010001010 => ? = 28
[1,0,1,1,0,0,1,1,0,0,1,0]
 => 101100110010 => 000101010100 => 001010101000 => ? = 24
[1,0,1,1,0,0,1,1,0,1,0,0]
 => 101100110100 => 010001010100 => 001010100010 => ? = 26
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000008
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 21%
Values
[1,0,1,0]
 => 1010 => [1,1,1,1] => [4] => 0
[1,1,0,0]
 => 1100 => [2,2] => [1,2,1] => 4
[1,0,1,0,1,0]
 => 101010 => [1,1,1,1,1,1] => [6] => 0
[1,0,1,1,0,0]
 => 101100 => [1,1,2,2] => [3,2,1] => 8
[1,1,0,0,1,0]
 => 110010 => [2,2,1,1] => [1,2,3] => 4
[1,1,0,1,0,0]
 => 110100 => [2,1,1,2] => [1,4,1] => 6
[1,1,1,0,0,0]
 => 111000 => [3,3] => [1,1,2,1,1] => 12
[1,0,1,0,1,0,1,0]
 => 10101010 => [1,1,1,1,1,1,1,1] => [8] => 0
[1,0,1,0,1,1,0,0]
 => 10101100 => [1,1,1,1,2,2] => [5,2,1] => 12
[1,0,1,1,0,0,1,0]
 => 10110010 => [1,1,2,2,1,1] => [3,2,3] => 8
[1,0,1,1,0,1,0,0]
 => 10110100 => [1,1,2,1,1,2] => [3,4,1] => 10
[1,0,1,1,1,0,0,0]
 => 10111000 => [1,1,3,3] => [3,1,2,1,1] => 20
[1,1,0,0,1,0,1,0]
 => 11001010 => [2,2,1,1,1,1] => [1,2,5] => 4
[1,1,0,0,1,1,0,0]
 => 11001100 => [2,2,2,2] => [1,2,2,2,1] => 16
[1,1,0,1,0,0,1,0]
 => 11010010 => [2,1,1,2,1,1] => [1,4,3] => 6
[1,1,0,1,0,1,0,0]
 => 11010100 => [2,1,1,1,1,2] => [1,6,1] => 8
[1,1,0,1,1,0,0,0]
 => 11011000 => [2,1,2,3] => [1,3,2,1,1] => 18
[1,1,1,0,0,0,1,0]
 => 11100010 => [3,3,1,1] => [1,1,2,1,3] => 12
[1,1,1,0,0,1,0,0]
 => 11100100 => [3,2,1,2] => [1,1,2,3,1] => 14
[1,1,1,0,1,0,0,0]
 => 11101000 => [3,1,1,3] => [1,1,4,1,1] => 16
[1,1,1,1,0,0,0,0]
 => 11110000 => [4,4] => [1,1,1,2,1,1,1] => 24
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => [1,1,1,1,1,1,1,1,1,1] => [10] => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => [1,1,1,1,1,1,2,2] => [7,2,1] => ? = 16
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => [1,1,1,1,2,2,1,1] => [5,2,3] => ? = 12
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => [1,1,1,1,2,1,1,2] => [5,4,1] => ? = 14
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => [1,1,1,1,3,3] => [5,1,2,1,1] => ? = 28
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => [1,1,2,2,1,1,1,1] => [3,2,5] => ? = 8
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => [1,1,2,2,2,2] => [3,2,2,2,1] => ? = 24
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => [1,1,2,1,1,2,1,1] => [3,4,3] => ? = 10
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => [1,1,2,1,1,1,1,2] => [3,6,1] => ? = 12
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => [1,1,2,1,2,3] => [3,3,2,1,1] => ? = 26
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => [1,1,3,3,1,1] => [3,1,2,1,3] => ? = 20
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => [1,1,3,2,1,2] => [3,1,2,3,1] => ? = 22
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => [1,1,3,1,1,3] => [3,1,4,1,1] => ? = 24
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => [1,1,4,4] => [3,1,1,2,1,1,1] => ? = 36
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => [2,2,1,1,1,1,1,1] => [1,2,7] => ? = 4
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => [2,2,1,1,2,2] => [1,2,4,2,1] => ? = 20
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => [2,2,2,2,1,1] => [1,2,2,2,3] => ? = 16
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => [2,2,2,1,1,2] => [1,2,2,4,1] => ? = 18
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => [2,2,3,3] => [1,2,2,1,2,1,1] => ? = 32
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => [2,1,1,2,1,1,1,1] => [1,4,5] => ? = 6
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => [2,1,1,2,2,2] => [1,4,2,2,1] => ? = 22
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => [2,1,1,1,1,2,1,1] => [1,6,3] => ? = 8
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => [2,1,1,1,1,1,1,2] => [1,8,1] => 10
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => [2,1,1,1,2,3] => [1,5,2,1,1] => ? = 24
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => [2,1,2,3,1,1] => [1,3,2,1,3] => ? = 18
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => [2,1,2,2,1,2] => [1,3,2,3,1] => ? = 20
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => [2,1,2,1,1,3] => [1,3,4,1,1] => ? = 22
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => [2,1,3,4] => [1,3,1,2,1,1,1] => ? = 34
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => [3,3,1,1,1,1] => [1,1,2,1,5] => ? = 12
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => [3,3,2,2] => [1,1,2,1,2,2,1] => ? = 28
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => [3,2,1,2,1,1] => [1,1,2,3,3] => ? = 14
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => [3,2,1,1,1,2] => [1,1,2,5,1] => ? = 16
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => [3,2,2,3] => [1,1,2,2,2,1,1] => ? = 30
[1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => [3,1,1,3,1,1] => [1,1,4,1,3] => ? = 16
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => [3,1,1,2,1,2] => [1,1,4,3,1] => ? = 18
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => [3,1,1,1,1,3] => [1,1,6,1,1] => ? = 20
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => [3,1,2,4] => [1,1,3,2,1,1,1] => ? = 32
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => [4,4,1,1] => [1,1,1,2,1,1,3] => ? = 24
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => [4,3,1,2] => [1,1,1,2,1,3,1] => ? = 26
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => [4,2,1,3] => [1,1,1,2,3,1,1] => ? = 28
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => [4,1,1,4] => [1,1,1,4,1,1,1] => ? = 30
[1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => [5,5] => [1,1,1,1,2,1,1,1,1] => ? = 40
[1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => [1,1,1,1,1,1,1,1,1,1,1,1] => [12] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => [1,1,1,1,1,1,1,1,2,2] => [9,2,1] => ? = 20
[1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => [1,1,1,1,1,1,2,2,1,1] => [7,2,3] => ? = 16
[1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => [1,1,1,1,1,1,2,1,1,2] => [7,4,1] => ? = 18
[1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => [1,1,1,1,1,1,3,3] => [7,1,2,1,1] => ? = 36
[1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => [1,1,1,1,2,2,1,1,1,1] => [5,2,5] => ? = 12
[1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => [1,1,1,1,2,2,2,2] => [5,2,2,2,1] => ? = 32
[1,0,1,0,1,1,0,1,0,0,1,0]
 => 101011010010 => [1,1,1,1,2,1,1,2,1,1] => [5,4,3] => ? = 14
[1,0,1,0,1,1,0,1,0,1,0,0]
 => 101011010100 => [1,1,1,1,2,1,1,1,1,2] => [5,6,1] => ? = 16
Description
The major index of the composition. The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].