Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000981
(load all 20 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000981: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 6
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => 4
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 8
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 5
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 5
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 6
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 8
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 5
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 5
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 4
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 6
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 8
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 5
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 5
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 4
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 6
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 3
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 3
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 3
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 4
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 8
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 5
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 3
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 3
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 5
Description
The length of the longest zigzag subpath. This is the length of the longest consecutive subpath that is a zigzag of the form $010...$ or of the form $101...$.
Matching statistic: St000983
(load all 9 compositions to match this statistic)
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000983: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 4
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 6
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 3
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 3
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 4
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 8
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 5
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 3
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 4
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 3
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 5
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 2
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 4
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 6
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 3
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 3
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 3
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 8
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 5
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 3
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 5
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 2
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 4
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 6
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 3
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 3
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 3
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 4
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 8
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 5
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 4
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 5
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => 4
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 6
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 3
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => 3
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => 3
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 4
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 8
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 5
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 3
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 3
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => 5
Description
The length of the longest alternating subword. This is the length of the longest consecutive subword of the form $010...$ or of the form $101...$.
Matching statistic: St000982
(load all 4 compositions to match this statistic)
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => 1010 => 0101 => 0000 => 4
[1,0,1,0,1,0]
 => 101010 => 010101 => 000000 => 6
[1,0,1,1,0,0]
 => 101100 => 001101 => 011000 => 3
[1,1,0,0,1,0]
 => 110010 => 010011 => 000110 => 3
[1,1,0,1,0,0]
 => 110100 => 001011 => 011110 => 4
[1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 00000000 => 8
[1,0,1,0,1,1,0,0]
 => 10101100 => 00110101 => 01100000 => 5
[1,0,1,1,0,0,1,0]
 => 10110010 => 01001101 => 00011000 => 3
[1,0,1,1,0,1,0,0]
 => 10110100 => 00101101 => 01111000 => 4
[1,0,1,1,1,0,0,0]
 => 10111000 => 00011101 => 01001000 => 3
[1,1,0,0,1,0,1,0]
 => 11001010 => 01010011 => 00000110 => 5
[1,1,0,0,1,1,0,0]
 => 11001100 => 00110011 => 01100110 => 2
[1,1,0,1,0,0,1,0]
 => 11010010 => 01001011 => 00011110 => 4
[1,1,0,1,0,1,0,0]
 => 11010100 => 00101011 => 01111110 => 6
[1,1,0,1,1,0,0,0]
 => 11011000 => 00011011 => 01001110 => 3
[1,1,1,0,0,0,1,0]
 => 11100010 => 01000111 => 00010010 => 3
[1,1,1,0,0,1,0,0]
 => 11100100 => 00100111 => 01110010 => 3
[1,1,1,0,1,0,0,0]
 => 11101000 => 00010111 => 01000010 => 4
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 0010101011 => 0111111110 => ? = 8
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 0001101011 => 0100111110 => ? = 5
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 0010011011 => 0111001110 => ? = 3
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 0001011011 => 0100001110 => ? = 4
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 0000111011 => 0101101110 => ? = 3
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 0010100111 => 0111110010 => ? = 5
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 0001100111 => 0100110010 => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 0010010111 => 0111000010 => ? = 4
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 0001010111 => 0100000010 => 6
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 0000110111 => 0101100010 => ? = 3
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 0010001111 => 0111011010 => ? = 3
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 0001001111 => 0100011010 => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 0000101111 => 0101111010 => ? = 4
[1,1,1,0,1,0,1,0,1,0,0,0]
 => 111010101000 => 000101010111 => 010000000010 => ? = 8
[1,1,1,0,1,0,1,1,0,0,0,0]
 => 111010110000 => 000011010111 => 010110000010 => ? = 5
[1,1,1,0,1,1,0,0,1,0,0,0]
 => 111011001000 => 000100110111 => 010001100010 => ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => 111011010000 => 000010110111 => 010111100010 => 4
[1,1,1,0,1,1,1,0,0,0,0,0]
 => 111011100000 => 000001110111 => 010100100010 => ? = 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => 111100101000 => 000101001111 => 010000011010 => ? = 5
[1,1,1,1,0,0,1,1,0,0,0,0]
 => 111100110000 => 000011001111 => 010110011010 => 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => 111101001000 => 000100101111 => 010001111010 => 4
[1,1,1,1,0,1,0,1,0,0,0,0]
 => 111101010000 => 000010101111 => 010111111010 => ? = 6
[1,1,1,1,0,1,1,0,0,0,0,0]
 => 111101100000 => 000001101111 => 010100111010 => ? = 3
[1,1,1,1,1,0,0,0,1,0,0,0]
 => 111110001000 => 000100011111 => 010001001010 => ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0]
 => 111110010000 => 000010011111 => 010111001010 => 3
[1,1,1,1,1,0,1,0,0,0,0,0]
 => 111110100000 => 000001011111 => 010100001010 => ? = 4
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 11110101010000 => 00001010101111 => 01011111111010 => ? = 8
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => 11110101100000 => 00000110101111 => 01010011111010 => ? = 5
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => 11110110010000 => 00001001101111 => 01011100111010 => ? = 3
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => 11110110100000 => 00000101101111 => 01010000111010 => ? = 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => 11110111000000 => 00000011101111 => 01010110111010 => ? = 3
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => 11111001010000 => 00001010011111 => 01011111001010 => ? = 5
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => 11111001100000 => 00000110011111 => 01010011001010 => ? = 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => 11111010010000 => 00001001011111 => 01011100001010 => ? = 4
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 11111010100000 => 00000101011111 => 01010000001010 => ? = 6
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => 11111011000000 => 00000011011111 => 01010110001010 => ? = 3
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 11111100010000 => 00001000111111 => 01011101101010 => ? = 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 11111100100000 => 00000100111111 => 01010001101010 => ? = 3
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 11111101000000 => 00000010111111 => 01010111101010 => ? = 4
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => 1111101010100000 => 0000010101011111 => ? => ? = 8
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => 1111101011000000 => 0000001101011111 => ? => ? = 5
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => 1111101100100000 => 0000010011011111 => ? => ? = 3
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => 1111101101000000 => 0000001011011111 => ? => ? = 4
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => 1111101110000000 => 0000000111011111 => ? => ? = 3
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => 1111110010100000 => 0000010100111111 => ? => ? = 5
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => 1111110011000000 => 0000001100111111 => ? => ? = 2
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => 1111110100100000 => 0000010010111111 => ? => ? = 4
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 1111110101000000 => 0000001010111111 => ? => ? = 6
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => 1111110110000000 => 0000000110111111 => ? => ? = 3
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => 1111111000100000 => 0000010001111111 => ? => ? = 3
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => 1111111001000000 => 0000001001111111 => ? => ? = 3
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => 1111111010000000 => 0000000101111111 => ? => ? = 4
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => 111111010101000000 => ? => ? => ? = 8
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => 111111010110000000 => 000000011010111111 => ? => ? = 5
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => 111111011001000000 => 000000100110111111 => ? => ? = 3
Description
The length of the longest constant subword.
Matching statistic: St000381
(load all 2 compositions to match this statistic)
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
 => 1010 => [1,1,1,1] => [4] => 4
[1,0,1,0,1,0]
 => 101010 => [1,1,1,1,1,1] => [6] => 6
[1,0,1,1,0,0]
 => 101100 => [1,1,2,2] => [3,2,1] => 3
[1,1,0,0,1,0]
 => 110010 => [2,2,1,1] => [1,2,3] => 3
[1,1,0,1,0,0]
 => 110100 => [2,1,1,2] => [1,4,1] => 4
[1,0,1,0,1,0,1,0]
 => 10101010 => [1,1,1,1,1,1,1,1] => [8] => 8
[1,0,1,0,1,1,0,0]
 => 10101100 => [1,1,1,1,2,2] => [5,2,1] => 5
[1,0,1,1,0,0,1,0]
 => 10110010 => [1,1,2,2,1,1] => [3,2,3] => 3
[1,0,1,1,0,1,0,0]
 => 10110100 => [1,1,2,1,1,2] => [3,4,1] => 4
[1,0,1,1,1,0,0,0]
 => 10111000 => [1,1,3,3] => [3,1,2,1,1] => 3
[1,1,0,0,1,0,1,0]
 => 11001010 => [2,2,1,1,1,1] => [1,2,5] => 5
[1,1,0,0,1,1,0,0]
 => 11001100 => [2,2,2,2] => [1,2,2,2,1] => 2
[1,1,0,1,0,0,1,0]
 => 11010010 => [2,1,1,2,1,1] => [1,4,3] => 4
[1,1,0,1,0,1,0,0]
 => 11010100 => [2,1,1,1,1,2] => [1,6,1] => 6
[1,1,0,1,1,0,0,0]
 => 11011000 => [2,1,2,3] => [1,3,2,1,1] => 3
[1,1,1,0,0,0,1,0]
 => 11100010 => [3,3,1,1] => [1,1,2,1,3] => 3
[1,1,1,0,0,1,0,0]
 => 11100100 => [3,2,1,2] => [1,1,2,3,1] => 3
[1,1,1,0,1,0,0,0]
 => 11101000 => [3,1,1,3] => [1,1,4,1,1] => 4
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => [2,1,1,1,1,1,1,2] => [1,8,1] => 8
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => [2,1,1,1,2,3] => [1,5,2,1,1] => ? = 5
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => [2,1,2,2,1,2] => [1,3,2,3,1] => ? = 3
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => [2,1,2,1,1,3] => [1,3,4,1,1] => ? = 4
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => [2,1,3,4] => [1,3,1,2,1,1,1] => ? = 3
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => [3,2,1,1,1,2] => [1,1,2,5,1] => ? = 5
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => [3,2,2,3] => [1,1,2,2,2,1,1] => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => [3,1,1,2,1,2] => [1,1,4,3,1] => ? = 4
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => [3,1,1,1,1,3] => [1,1,6,1,1] => ? = 6
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => [3,1,2,4] => [1,1,3,2,1,1,1] => ? = 3
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => [4,3,1,2] => [1,1,1,2,1,3,1] => ? = 3
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => [4,2,1,3] => [1,1,1,2,3,1,1] => ? = 3
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => [4,1,1,4] => [1,1,1,4,1,1,1] => ? = 4
[1,1,1,0,1,0,1,0,1,0,0,0]
 => 111010101000 => [3,1,1,1,1,1,1,3] => [1,1,8,1,1] => ? = 8
[1,1,1,0,1,0,1,1,0,0,0,0]
 => 111010110000 => [3,1,1,1,2,4] => [1,1,5,2,1,1,1] => ? = 5
[1,1,1,0,1,1,0,0,1,0,0,0]
 => 111011001000 => [3,1,2,2,1,3] => [1,1,3,2,3,1,1] => ? = 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => 111011010000 => [3,1,2,1,1,4] => [1,1,3,4,1,1,1] => ? = 4
[1,1,1,0,1,1,1,0,0,0,0,0]
 => 111011100000 => [3,1,3,5] => [1,1,3,1,2,1,1,1,1] => ? = 3
[1,1,1,1,0,0,1,0,1,0,0,0]
 => 111100101000 => [4,2,1,1,1,3] => [1,1,1,2,5,1,1] => ? = 5
[1,1,1,1,0,0,1,1,0,0,0,0]
 => 111100110000 => [4,2,2,4] => [1,1,1,2,2,2,1,1,1] => ? = 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => 111101001000 => [4,1,1,2,1,3] => [1,1,1,4,3,1,1] => ? = 4
[1,1,1,1,0,1,0,1,0,0,0,0]
 => 111101010000 => [4,1,1,1,1,4] => [1,1,1,6,1,1,1] => ? = 6
[1,1,1,1,0,1,1,0,0,0,0,0]
 => 111101100000 => [4,1,2,5] => [1,1,1,3,2,1,1,1,1] => ? = 3
[1,1,1,1,1,0,0,0,1,0,0,0]
 => 111110001000 => [5,3,1,3] => [1,1,1,1,2,1,3,1,1] => ? = 3
[1,1,1,1,1,0,0,1,0,0,0,0]
 => 111110010000 => [5,2,1,4] => [1,1,1,1,2,3,1,1,1] => ? = 3
[1,1,1,1,1,0,1,0,0,0,0,0]
 => 111110100000 => [5,1,1,5] => [1,1,1,1,4,1,1,1,1] => ? = 4
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 11110101010000 => [4,1,1,1,1,1,1,4] => [1,1,1,8,1,1,1] => ? = 8
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => 11110101100000 => [4,1,1,1,2,5] => [1,1,1,5,2,1,1,1,1] => ? = 5
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => 11110110010000 => [4,1,2,2,1,4] => [1,1,1,3,2,3,1,1,1] => ? = 3
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => 11110110100000 => [4,1,2,1,1,5] => [1,1,1,3,4,1,1,1,1] => ? = 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => 11110111000000 => [4,1,3,6] => [1,1,1,3,1,2,1,1,1,1,1] => ? = 3
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => 11111001010000 => [5,2,1,1,1,4] => [1,1,1,1,2,5,1,1,1] => ? = 5
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => 11111001100000 => [5,2,2,5] => [1,1,1,1,2,2,2,1,1,1,1] => ? = 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => 11111010010000 => [5,1,1,2,1,4] => [1,1,1,1,4,3,1,1,1] => ? = 4
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => 11111010100000 => [5,1,1,1,1,5] => [1,1,1,1,6,1,1,1,1] => ? = 6
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => 11111011000000 => [5,1,2,6] => [1,1,1,1,3,2,1,1,1,1,1] => ? = 3
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 11111100010000 => [6,3,1,4] => [1,1,1,1,1,2,1,3,1,1,1] => ? = 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 11111100100000 => [6,2,1,5] => [1,1,1,1,1,2,3,1,1,1,1] => ? = 3
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 11111101000000 => [6,1,1,6] => [1,1,1,1,1,4,1,1,1,1,1] => ? = 4
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => 1111101010100000 => [5,1,1,1,1,1,1,5] => ? => ? = 8
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => 1111101011000000 => [5,1,1,1,2,6] => ? => ? = 5
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => 1111101100100000 => [5,1,2,2,1,5] => ? => ? = 3
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => 1111101101000000 => ? => ? => ? = 4
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => 1111101110000000 => [5,1,3,7] => ? => ? = 3
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => 1111110010100000 => [6,2,1,1,1,5] => ? => ? = 5
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => 1111110011000000 => ? => ? => ? = 2
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => 1111110100100000 => ? => ? => ? = 4
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => 1111110101000000 => [6,1,1,1,1,6] => ? => ? = 6
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => 1111110110000000 => [6,1,2,7] => ? => ? = 3
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => 1111111000100000 => [7,3,1,5] => ? => ? = 3
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => 1111111001000000 => [7,2,1,6] => ? => ? = 3
Description
The largest part of an integer composition.
Matching statistic: St000422
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[1,0,1,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2 = 4 - 2
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 4 = 6 - 2
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => ? = 3 - 2
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? = 3 - 2
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => 2 = 4 - 2
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6 = 8 - 2
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 - 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 - 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 5 - 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 4 = 6 - 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? = 3 - 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 3 - 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 3 - 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 4 - 2
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6 = 8 - 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ? = 3 - 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3 - 2
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 2
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? = 2 - 2
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 2
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => 4 = 6 - 2
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? = 3 - 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3 - 2
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? = 3 - 2
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 4 - 2
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 8 - 2
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,2,1,5,6] => [4,2,1,3,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 2
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [4,2,3,1,5,6] => [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ? = 3 - 2
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,4,1,5,6] => [4,1,3,2,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,1,5,6] => [4,1,2,3,5,6] => ([(2,5),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,3,1,2,5,6] => [2,4,3,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 2
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [3,4,1,2,5,6] => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ? = 2 - 2
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,2,1,3,5,6] => [3,4,2,1,5,6] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 2
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,4,5,6] => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => 4 = 6 - 2
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6] => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [4,1,2,3,5,6] => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [3,1,2,4,5,6] => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ? = 3 - 2
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(4,5)],6)
 => 2 = 4 - 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6 = 8 - 2
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [3,4,2,1,5,6,7] => [4,2,1,3,5,6,7] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 - 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [4,2,3,1,5,6,7] => [3,1,4,2,5,6,7] => ([(3,6),(4,5),(5,6)],7)
 => ? = 3 - 2
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [3,2,4,1,5,6,7] => [4,1,3,2,5,6,7] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,3,4,1,5,6,7] => [4,1,2,3,5,6,7] => ([(3,6),(4,6),(5,6)],7)
 => ? = 3 - 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1,2,5,6,7] => [2,4,3,1,5,6,7] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 5 - 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [3,4,1,2,5,6,7] => [2,4,1,3,5,6,7] => ([(3,6),(4,5),(5,6)],7)
 => ? = 2 - 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1,3,5,6,7] => [3,4,2,1,5,6,7] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7)
 => 4 = 6 - 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => [3,1,2,4,5,6,7] => ([(4,6),(5,6)],7)
 => ? = 3 - 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => ([(3,6),(4,6),(5,6)],7)
 => ? = 3 - 2
[1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => ([(4,6),(5,6)],7)
 => ? = 3 - 2
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ([(5,6)],7)
 => 2 = 4 - 2
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 - 2
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [3,4,2,1,5,6,7,8] => ? => ?
 => ? = 5 - 2
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [4,2,3,1,5,6,7,8] => ? => ?
 => ? = 3 - 2
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [3,2,4,1,5,6,7,8] => [4,1,3,2,5,6,7,8] => ([(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 - 2
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [2,3,4,1,5,6,7,8] => [4,1,2,3,5,6,7,8] => ([(4,7),(5,7),(6,7)],8)
 => ? = 3 - 2
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [4,3,1,2,5,6,7,8] => ? => ?
 => ? = 5 - 2
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => [3,4,1,2,5,6,7,8] => [2,4,1,3,5,6,7,8] => ([(4,7),(5,6),(6,7)],8)
 => ? = 2 - 2
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [4,2,1,3,5,6,7,8] => ? => ?
 => ? = 4 - 2
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001491
Mapping
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 17%
Values
[1,0,1,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 0 = 4 - 4
[1,0,1,0,1,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 6 - 4
[1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 3 - 4
[1,1,0,0,1,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 3 - 4
[1,1,0,1,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 0 = 4 - 4
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 8 - 4
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 4
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => ? = 3 - 4
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 4 - 4
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 3 - 4
[1,1,0,0,1,0,1,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => ? = 5 - 4
[1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 2 - 4
[1,1,0,1,0,0,1,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 4
[1,1,0,1,0,1,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 6 - 4
[1,1,0,1,1,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 3 - 4
[1,1,1,0,0,0,1,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 3 - 4
[1,1,1,0,0,1,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 3 - 4
[1,1,1,0,1,0,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 0 = 4 - 4
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 8 - 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 4
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => ? = 3 - 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 4 - 4
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 3 - 4
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => ? = 5 - 4
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 2 - 4
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 4
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 6 - 4
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 3 - 4
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 3 - 4
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 3 - 4
[1,1,1,1,0,1,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 0 = 4 - 4
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 8 - 4
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 4
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => ? = 3 - 4
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 4 - 4
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 3 - 4
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => ? = 5 - 4
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 2 - 4
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 4
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 6 - 4
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 3 - 4
[1,1,1,1,1,0,0,0,1,0,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 3 - 4
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [2]
 => [1,1,0,0,1,0]
 => 110010 => ? = 3 - 4
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 0 = 4 - 4
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => ? = 8 - 4
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => ? = 5 - 4
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => ? = 3 - 4
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 4 - 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 3 - 4
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 11001010 => ? = 5 - 4
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 2 - 4
[1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 4 - 4
[1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => ? = 6 - 4
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => ? = 3 - 4
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 11100010 => ? = 3 - 4
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 0 = 4 - 4
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 0 = 4 - 4
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 0 = 4 - 4
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 0 = 4 - 4
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [1]
 => [1,0,1,0]
 => 1010 => 0 = 4 - 4
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.