Identifier
Values
[1] => [1] => 10 => 11 => 1
[2] => [1,1] => 110 => 111 => 1
[1,1] => [2] => 100 => 011 => 2
[3] => [1,1,1] => 1110 => 1111 => 1
[2,1] => [2,1] => 1010 => 1101 => 1
[1,1,1] => [3] => 1000 => 0011 => 3
[4] => [1,1,1,1] => 11110 => 11111 => 1
[3,1] => [2,1,1] => 10110 => 11011 => 1
[2,2] => [2,2] => 1100 => 0111 => 2
[2,1,1] => [3,1] => 10010 => 01101 => 2
[1,1,1,1] => [4] => 10000 => 00011 => 4
[5] => [1,1,1,1,1] => 111110 => 111111 => 1
[4,1] => [2,1,1,1] => 101110 => 110111 => 1
[3,2] => [2,2,1] => 11010 => 11101 => 1
[3,1,1] => [3,1,1] => 100110 => 011011 => 2
[2,2,1] => [3,2] => 10100 => 11001 => 1
[2,1,1,1] => [4,1] => 100010 => 001101 => 3
[1,1,1,1,1] => [5] => 100000 => 000011 => 5
[6] => [1,1,1,1,1,1] => 1111110 => 1111111 => 1
[5,1] => [2,1,1,1,1] => 1011110 => 1101111 => 1
[4,2] => [2,2,1,1] => 110110 => 111011 => 1
[4,1,1] => [3,1,1,1] => 1001110 => 0110111 => 2
[3,3] => [2,2,2] => 11100 => 01111 => 2
[3,2,1] => [3,2,1] => 101010 => 110101 => 1
[3,1,1,1] => [4,1,1] => 1000110 => 0011011 => 3
[2,2,2] => [3,3] => 11000 => 00111 => 3
[2,2,1,1] => [4,2] => 100100 => 011001 => 2
[2,1,1,1,1] => [5,1] => 1000010 => 0001101 => 4
[1,1,1,1,1,1] => [6] => 1000000 => 0000011 => 6
[7] => [1,1,1,1,1,1,1] => 11111110 => 11111111 => 1
[6,1] => [2,1,1,1,1,1] => 10111110 => 11011111 => 1
[5,2] => [2,2,1,1,1] => 1101110 => 1110111 => 1
[5,1,1] => [3,1,1,1,1] => 10011110 => 01101111 => 2
[4,3] => [2,2,2,1] => 111010 => 111101 => 1
[4,2,1] => [3,2,1,1] => 1010110 => 1101011 => 1
[4,1,1,1] => [4,1,1,1] => 10001110 => 00110111 => 3
[3,3,1] => [3,2,2] => 101100 => 110011 => 1
[3,2,2] => [3,3,1] => 110010 => 011101 => 2
[3,2,1,1] => [4,2,1] => 1001010 => 0110101 => 2
[3,1,1,1,1] => [5,1,1] => 10000110 => 00011011 => 4
[2,2,2,1] => [4,3] => 101000 => 110001 => 1
[2,2,1,1,1] => [5,2] => 1000100 => 0011001 => 3
[2,1,1,1,1,1] => [6,1] => 10000010 => 00001101 => 5
[1,1,1,1,1,1,1] => [7] => 10000000 => 00000011 => 7
[8] => [1,1,1,1,1,1,1,1] => 111111110 => 111111111 => 1
[7,1] => [2,1,1,1,1,1,1] => 101111110 => 110111111 => 1
[6,2] => [2,2,1,1,1,1] => 11011110 => 11101111 => 1
[6,1,1] => [3,1,1,1,1,1] => 100111110 => 011011111 => 2
[5,3] => [2,2,2,1,1] => 1110110 => 1111011 => 1
[5,2,1] => [3,2,1,1,1] => 10101110 => 11010111 => 1
[5,1,1,1] => [4,1,1,1,1] => 100011110 => 001101111 => 3
[4,4] => [2,2,2,2] => 111100 => 011111 => 2
[4,3,1] => [3,2,2,1] => 1011010 => 1101101 => 1
[4,2,2] => [3,3,1,1] => 1100110 => 0111011 => 2
[4,2,1,1] => [4,2,1,1] => 10010110 => 01101011 => 2
[4,1,1,1,1] => [5,1,1,1] => 100001110 => 000110111 => 4
[3,3,2] => [3,3,2] => 110100 => 111001 => 1
[3,3,1,1] => [4,2,2] => 1001100 => 0110011 => 2
[3,2,2,1] => [4,3,1] => 1010010 => 1100101 => 1
[3,2,1,1,1] => [5,2,1] => 10001010 => 00110101 => 3
[3,1,1,1,1,1] => [6,1,1] => 100000110 => 000011011 => 5
[2,2,2,2] => [4,4] => 110000 => 000111 => 4
[2,2,2,1,1] => [5,3] => 1001000 => 0110001 => 2
[2,2,1,1,1,1] => [6,2] => 10000100 => 00011001 => 4
[2,1,1,1,1,1,1] => [7,1] => 100000010 => 000001101 => 6
[1,1,1,1,1,1,1,1] => [8] => 100000000 => 000000011 => 8
[7,2] => [2,2,1,1,1,1,1] => 110111110 => 111011111 => 1
[7,1,1] => [3,1,1,1,1,1,1] => 1001111110 => 0110111111 => 2
[6,3] => [2,2,2,1,1,1] => 11101110 => 11110111 => 1
[6,2,1] => [3,2,1,1,1,1] => 101011110 => 110101111 => 1
[5,4] => [2,2,2,2,1] => 1111010 => 1111101 => 1
[5,3,1] => [3,2,2,1,1] => 10110110 => 11011011 => 1
[5,2,2] => [3,3,1,1,1] => 11001110 => 01110111 => 2
[5,2,1,1] => [4,2,1,1,1] => 100101110 => 011010111 => 2
[5,1,1,1,1] => [5,1,1,1,1] => 1000011110 => 0001101111 => 4
[4,4,1] => [3,2,2,2] => 1011100 => 1100111 => 1
[4,3,2] => [3,3,2,1] => 1101010 => 1110101 => 1
[4,3,1,1] => [4,2,2,1] => 10011010 => 01101101 => 2
[4,2,2,1] => [4,3,1,1] => 10100110 => 11001011 => 1
[4,2,1,1,1] => [5,2,1,1] => 100010110 => 001101011 => 3
[4,1,1,1,1,1] => [6,1,1,1] => 1000001110 => 0000110111 => 5
[3,3,3] => [3,3,3] => 111000 => 001111 => 3
[3,3,2,1] => [4,3,2] => 1010100 => 1101001 => 1
[3,3,1,1,1] => [5,2,2] => 10001100 => 00110011 => 3
[3,2,2,2] => [4,4,1] => 1100010 => 0011101 => 3
[3,2,2,1,1] => [5,3,1] => 10010010 => 01100101 => 2
[3,2,1,1,1,1] => [6,2,1] => 100001010 => 000110101 => 4
[3,1,1,1,1,1,1] => [7,1,1] => 1000000110 => 0000011011 => 6
[2,2,2,2,1] => [5,4] => 1010000 => 1100001 => 1
[2,2,2,1,1,1] => [6,3] => 10001000 => 00110001 => 3
[2,2,1,1,1,1,1] => [7,2] => 100000100 => 000011001 => 5
[2,1,1,1,1,1,1,1] => [8,1] => 1000000010 => 0000001101 => 7
[1,1,1,1,1,1,1,1,1] => [9] => 1000000000 => 0000000011 => 9
[7,3] => [2,2,2,1,1,1,1] => 111011110 => 111101111 => 1
[6,4] => [2,2,2,2,1,1] => 11110110 => 11111011 => 1
[6,3,1] => [3,2,2,1,1,1] => 101101110 => 110110111 => 1
[6,2,2] => [3,3,1,1,1,1] => 110011110 => 011101111 => 2
[5,5] => [2,2,2,2,2] => 1111100 => 0111111 => 2
[5,4,1] => [3,2,2,2,1] => 10111010 => 11011101 => 1
[5,3,2] => [3,3,2,1,1] => 11010110 => 11101011 => 1
[5,3,1,1] => [4,2,2,1,1] => 100110110 => 011011011 => 2
>>> Load all 263 entries. <<<
[5,2,2,1] => [4,3,1,1,1] => 101001110 => 110010111 => 1
[4,4,2] => [3,3,2,2] => 1101100 => 1110011 => 1
[4,4,1,1] => [4,2,2,2] => 10011100 => 01100111 => 2
[4,3,3] => [3,3,3,1] => 1110010 => 0111101 => 2
[4,3,2,1] => [4,3,2,1] => 10101010 => 11010101 => 1
[4,3,1,1,1] => [5,2,2,1] => 100011010 => 001101101 => 3
[4,2,2,2] => [4,4,1,1] => 11000110 => 00111011 => 3
[4,2,2,1,1] => [5,3,1,1] => 100100110 => 011001011 => 2
[4,2,1,1,1,1] => [6,2,1,1] => 1000010110 => 0001101011 => 4
[3,3,3,1] => [4,3,3] => 1011000 => 1100011 => 1
[3,3,2,2] => [4,4,2] => 1100100 => 0111001 => 2
[3,3,2,1,1] => [5,3,2] => 10010100 => 01101001 => 2
[3,3,1,1,1,1] => [6,2,2] => 100001100 => 000110011 => 4
[3,2,2,2,1] => [5,4,1] => 10100010 => 11000101 => 1
[3,2,2,1,1,1] => [6,3,1] => 100010010 => 001100101 => 3
[3,2,1,1,1,1,1] => [7,2,1] => 1000001010 => 0000110101 => 5
[2,2,2,2,2] => [5,5] => 1100000 => 0000111 => 5
[2,2,2,2,1,1] => [6,4] => 10010000 => 01100001 => 2
[2,2,2,1,1,1,1] => [7,3] => 100001000 => 000110001 => 4
[2,2,1,1,1,1,1,1] => [8,2] => 1000000100 => 0000011001 => 6
[2,1,1,1,1,1,1,1,1] => [9,1] => 10000000010 => 00000001101 => 8
[7,4] => [2,2,2,2,1,1,1] => 111101110 => 111110111 => 1
[6,5] => [2,2,2,2,2,1] => 11111010 => 11111101 => 1
[6,4,1] => [3,2,2,2,1,1] => 101110110 => 110111011 => 1
[6,3,2] => [3,3,2,1,1,1] => 110101110 => 111010111 => 1
[5,5,1] => [3,2,2,2,2] => 10111100 => 11001111 => 1
[5,4,2] => [3,3,2,2,1] => 11011010 => 11101101 => 1
[5,4,1,1] => [4,2,2,2,1] => 100111010 => 011011101 => 2
[5,3,3] => [3,3,3,1,1] => 11100110 => 01111011 => 2
[5,3,2,1] => [4,3,2,1,1] => 101010110 => 110101011 => 1
[5,2,2,2] => [4,4,1,1,1] => 110001110 => 001110111 => 3
[4,4,3] => [3,3,3,2] => 1110100 => 1111001 => 1
[4,4,2,1] => [4,3,2,2] => 10101100 => 11010011 => 1
[4,4,1,1,1] => [5,2,2,2] => 100011100 => 001100111 => 3
[4,3,3,1] => [4,3,3,1] => 10110010 => 11001101 => 1
[4,3,2,2] => [4,4,2,1] => 11001010 => 01110101 => 2
[4,3,2,1,1] => [5,3,2,1] => 100101010 => 011010101 => 2
[4,2,2,2,1] => [5,4,1,1] => 101000110 => 110001011 => 1
[3,3,3,2] => [4,4,3] => 1101000 => 1110001 => 1
[3,3,3,1,1] => [5,3,3] => 10011000 => 01100011 => 2
[3,3,2,2,1] => [5,4,2] => 10100100 => 11001001 => 1
[3,3,2,1,1,1] => [6,3,2] => 100010100 => 001101001 => 3
[3,2,2,2,2] => [5,5,1] => 11000010 => 00011101 => 4
[3,2,2,2,1,1] => [6,4,1] => 100100010 => 011000101 => 2
[2,2,2,2,2,1] => [6,5] => 10100000 => 11000001 => 1
[2,2,2,2,1,1,1] => [7,4] => 100010000 => 001100001 => 3
[7,5] => [2,2,2,2,2,1,1] => 111110110 => 111111011 => 1
[6,6] => [2,2,2,2,2,2] => 11111100 => 01111111 => 2
[6,5,1] => [3,2,2,2,2,1] => 101111010 => 110111101 => 1
[6,4,2] => [3,3,2,2,1,1] => 110110110 => 111011011 => 1
[6,3,3] => [3,3,3,1,1,1] => 111001110 => 011110111 => 2
[5,5,2] => [3,3,2,2,2] => 11011100 => 11100111 => 1
[5,5,1,1] => [4,2,2,2,2] => 100111100 => 011001111 => 2
[5,4,3] => [3,3,3,2,1] => 11101010 => 11110101 => 1
[5,4,2,1] => [4,3,2,2,1] => 101011010 => 110101101 => 1
[5,3,3,1] => [4,3,3,1,1] => 101100110 => 110011011 => 1
[5,3,2,2] => [4,4,2,1,1] => 110010110 => 011101011 => 2
[4,4,4] => [3,3,3,3] => 1111000 => 0011111 => 3
[4,4,3,1] => [4,3,3,2] => 10110100 => 11011001 => 1
[4,4,2,2] => [4,4,2,2] => 11001100 => 01110011 => 2
[4,4,2,1,1] => [5,3,2,2] => 100101100 => 011010011 => 2
[4,3,3,2] => [4,4,3,1] => 11010010 => 11100101 => 1
[4,3,3,1,1] => [5,3,3,1] => 100110010 => 011001101 => 2
[4,3,2,2,1] => [5,4,2,1] => 101001010 => 110010101 => 1
[4,2,2,2,2] => [5,5,1,1] => 110000110 => 000111011 => 4
[3,3,3,3] => [4,4,4] => 1110000 => 0001111 => 4
[3,3,3,2,1] => [5,4,3] => 10101000 => 11010001 => 1
[3,3,3,1,1,1] => [6,3,3] => 100011000 => 001100011 => 3
[3,3,2,2,2] => [5,5,2] => 11000100 => 00111001 => 3
[3,3,2,2,1,1] => [6,4,2] => 100100100 => 011001001 => 2
[3,2,2,2,2,1] => [6,5,1] => 101000010 => 110000101 => 1
[2,2,2,2,2,2] => [6,6] => 11000000 => 00000111 => 6
[2,2,2,2,2,1,1] => [7,5] => 100100000 => 011000001 => 2
[7,6] => [2,2,2,2,2,2,1] => 111111010 => 111111101 => 1
[6,6,1] => [3,2,2,2,2,2] => 101111100 => 110011111 => 1
[6,5,2] => [3,3,2,2,2,1] => 110111010 => 111011101 => 1
[6,4,3] => [3,3,3,2,1,1] => 111010110 => 111101011 => 1
[5,5,3] => [3,3,3,2,2] => 11101100 => 11110011 => 1
[5,5,2,1] => [4,3,2,2,2] => 101011100 => 110100111 => 1
[5,4,4] => [3,3,3,3,1] => 11110010 => 01111101 => 2
[5,4,3,1] => [4,3,3,2,1] => 101101010 => 110110101 => 1
[5,4,2,2] => [4,4,2,2,1] => 110011010 => 011101101 => 2
[5,3,3,2] => [4,4,3,1,1] => 110100110 => 111001011 => 1
[4,4,4,1] => [4,3,3,3] => 10111000 => 11000111 => 1
[4,4,3,2] => [4,4,3,2] => 11010100 => 11101001 => 1
[4,4,3,1,1] => [5,3,3,2] => 100110100 => 011011001 => 2
[4,4,2,2,1] => [5,4,2,2] => 101001100 => 110010011 => 1
[4,3,3,3] => [4,4,4,1] => 11100010 => 00111101 => 3
[4,3,3,2,1] => [5,4,3,1] => 101010010 => 110100101 => 1
[4,3,2,2,2] => [5,5,2,1] => 110001010 => 001110101 => 3
[3,3,3,3,1] => [5,4,4] => 10110000 => 11000011 => 1
[3,3,3,2,2] => [5,5,3] => 11001000 => 01110001 => 2
[3,3,3,2,1,1] => [6,4,3] => 100101000 => 011010001 => 2
[3,3,2,2,2,1] => [6,5,2] => 101000100 => 110001001 => 1
[3,2,2,2,2,2] => [6,6,1] => 110000010 => 000011101 => 5
[2,2,2,2,2,2,1] => [7,6] => 101000000 => 110000001 => 1
[7,7] => [2,2,2,2,2,2,2] => 111111100 => 011111111 => 2
[6,6,2] => [3,3,2,2,2,2] => 110111100 => 111001111 => 1
[6,5,3] => [3,3,3,2,2,1] => 111011010 => 111101101 => 1
[6,4,4] => [3,3,3,3,1,1] => 111100110 => 011111011 => 2
[5,5,4] => [3,3,3,3,2] => 11110100 => 11111001 => 1
[5,5,3,1] => [4,3,3,2,2] => 101101100 => 110110011 => 1
[5,5,2,2] => [4,4,2,2,2] => 110011100 => 011100111 => 2
[5,4,4,1] => [4,3,3,3,1] => 101110010 => 110011101 => 1
[5,4,3,2] => [4,4,3,2,1] => 110101010 => 111010101 => 1
[5,3,3,3] => [4,4,4,1,1] => 111000110 => 001111011 => 3
[4,4,4,2] => [4,4,3,3] => 11011000 => 11100011 => 1
[4,4,4,1,1] => [5,3,3,3] => 100111000 => 011000111 => 2
[4,4,3,3] => [4,4,4,2] => 11100100 => 01111001 => 2
[4,4,3,2,1] => [5,4,3,2] => 101010100 => 110101001 => 1
[4,4,2,2,2] => [5,5,2,2] => 110001100 => 001110011 => 3
[4,3,3,3,1] => [5,4,4,1] => 101100010 => 110001101 => 1
[4,3,3,2,2] => [5,5,3,1] => 110010010 => 011100101 => 2
[3,3,3,3,2] => [5,5,4] => 11010000 => 11100001 => 1
[3,3,3,3,1,1] => [6,4,4] => 100110000 => 011000011 => 2
[3,3,3,2,2,1] => [6,5,3] => 101001000 => 110010001 => 1
[3,3,2,2,2,2] => [6,6,2] => 110000100 => 000111001 => 4
[2,2,2,2,2,2,2] => [7,7] => 110000000 => 000000111 => 7
[6,6,3] => [3,3,3,2,2,2] => 111011100 => 111100111 => 1
[6,5,4] => [3,3,3,3,2,1] => 111101010 => 111110101 => 1
[5,5,5] => [3,3,3,3,3] => 11111000 => 00111111 => 3
[5,5,4,1] => [4,3,3,3,2] => 101110100 => 110111001 => 1
[5,5,3,2] => [4,4,3,2,2] => 110101100 => 111010011 => 1
[5,4,4,2] => [4,4,3,3,1] => 110110010 => 111001101 => 1
[5,4,3,3] => [4,4,4,2,1] => 111001010 => 011110101 => 2
[4,4,4,3] => [4,4,4,3] => 11101000 => 11110001 => 1
[4,4,4,2,1] => [5,4,3,3] => 101011000 => 110100011 => 1
[4,4,3,3,1] => [5,4,4,2] => 101100100 => 110011001 => 1
[4,4,3,2,2] => [5,5,3,2] => 110010100 => 011101001 => 2
[4,3,3,3,2] => [5,5,4,1] => 110100010 => 111000101 => 1
[3,3,3,3,3] => [5,5,5] => 11100000 => 00001111 => 5
[3,3,3,3,2,1] => [6,5,4] => 101010000 => 110100001 => 1
[3,3,3,2,2,2] => [6,6,3] => 110001000 => 001110001 => 3
[6,6,4] => [3,3,3,3,2,2] => 111101100 => 111110011 => 1
[6,5,5] => [3,3,3,3,3,1] => 111110010 => 011111101 => 2
[5,5,5,1] => [4,3,3,3,3] => 101111000 => 110001111 => 1
[5,5,4,2] => [4,4,3,3,2] => 110110100 => 111011001 => 1
[5,5,3,3] => [4,4,4,2,2] => 111001100 => 011110011 => 2
[5,4,4,3] => [4,4,4,3,1] => 111010010 => 111100101 => 1
[4,4,4,4] => [4,4,4,4] => 11110000 => 00011111 => 4
[4,4,4,3,1] => [5,4,4,3] => 101101000 => 110110001 => 1
[4,4,4,2,2] => [5,5,3,3] => 110011000 => 011100011 => 2
[4,4,3,3,2] => [5,5,4,2] => 110100100 => 111001001 => 1
[4,3,3,3,3] => [5,5,5,1] => 111000010 => 000111101 => 4
[3,3,3,3,3,1] => [6,5,5] => 101100000 => 110000011 => 1
[3,3,3,3,2,2] => [6,6,4] => 110010000 => 011100001 => 2
[6,6,5] => [3,3,3,3,3,2] => 111110100 => 111111001 => 1
[5,5,5,2] => [4,4,3,3,3] => 110111000 => 111000111 => 1
[5,5,4,3] => [4,4,4,3,2] => 111010100 => 111101001 => 1
[5,4,4,4] => [4,4,4,4,1] => 111100010 => 001111101 => 3
[4,4,4,4,1] => [5,4,4,4] => 101110000 => 110000111 => 1
[4,4,4,3,2] => [5,5,4,3] => 110101000 => 111010001 => 1
[4,4,3,3,3] => [5,5,5,2] => 111000100 => 001111001 => 3
[3,3,3,3,3,2] => [6,6,5] => 110100000 => 111000001 => 1
[3,3,3,2,2,2,2] => [7,7,3] => 1100001000 => 0001110001 => 4
[5,5,4,4] => [4,4,4,4,2] => 111100100 => 011111001 => 2
[3,3,3,3,3,3] => [6,6,6] => 111000000 => 000001111 => 6
[4,4,4,4,4] => [5,5,5,5] => 111100000 => 000011111 => 5
[5,5,5,5] => [4,4,4,4,4] => 111110000 => 000111111 => 4
[6,6,6] => [3,3,3,3,3,3] => 111111000 => 001111111 => 3
[5,5,5,5,5] => [5,5,5,5,5] => 1111100000 => 0000111111 => 5
[5,4,4,4,4] => [5,5,5,5,1] => 1111000010 => 0001111101 => 4
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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.
Map
path rowmotion
Description
Return the rowmotion of the binary word, regarded as a lattice path.
Consider the binary word of length $n$ as a lattice path with $n$ steps, where a 1 corresponds to an up step and a 0 corresponds to a down step.
This map returns the path whose peaks are the valleys of the original path with an up step appended.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.