Identifier
Values
0 => [2] => 1
1 => [1,1] => 1
00 => [3] => 1
01 => [2,1] => 1
10 => [1,2] => 1
11 => [1,1,1] => 1
000 => [4] => 1
001 => [3,1] => 1
010 => [2,2] => 2
011 => [2,1,1] => 1
100 => [1,3] => 2
101 => [1,2,1] => 1
110 => [1,1,2] => 1
111 => [1,1,1,1] => 1
0000 => [5] => 1
0001 => [4,1] => 1
0010 => [3,2] => 3
0011 => [3,1,1] => 1
0100 => [2,3] => 2
0101 => [2,2,1] => 2
0110 => [2,1,2] => 1
0111 => [2,1,1,1] => 1
1000 => [1,4] => 3
1001 => [1,3,1] => 2
1010 => [1,2,2] => 2
1011 => [1,2,1,1] => 1
1100 => [1,1,3] => 3
1101 => [1,1,2,1] => 1
1110 => [1,1,1,2] => 1
1111 => [1,1,1,1,1] => 1
00000 => [6] => 1
00001 => [5,1] => 1
00010 => [4,2] => 4
00011 => [4,1,1] => 1
00100 => [3,3] => 5
00101 => [3,2,1] => 3
00110 => [3,1,2] => 1
00111 => [3,1,1,1] => 1
01000 => [2,4] => 5
01001 => [2,3,1] => 2
01010 => [2,2,2] => 5
01011 => [2,2,1,1] => 2
01100 => [2,1,3] => 3
01101 => [2,1,2,1] => 1
01110 => [2,1,1,2] => 1
01111 => [2,1,1,1,1] => 1
10000 => [1,5] => 4
10001 => [1,4,1] => 3
10010 => [1,3,2] => 5
10011 => [1,3,1,1] => 2
10100 => [1,2,3] => 2
10101 => [1,2,2,1] => 2
10110 => [1,2,1,2] => 1
10111 => [1,2,1,1,1] => 1
11000 => [1,1,4] => 6
11001 => [1,1,3,1] => 3
11010 => [1,1,2,2] => 2
11011 => [1,1,2,1,1] => 1
11100 => [1,1,1,3] => 4
11101 => [1,1,1,2,1] => 1
11110 => [1,1,1,1,2] => 1
11111 => [1,1,1,1,1,1] => 1
000000 => [7] => 1
000001 => [6,1] => 1
000010 => [5,2] => 5
000011 => [5,1,1] => 1
000100 => [4,3] => 9
000101 => [4,2,1] => 4
000110 => [4,1,2] => 1
000111 => [4,1,1,1] => 1
001000 => [3,4] => 5
001001 => [3,3,1] => 5
001010 => [3,2,2] => 9
001011 => [3,2,1,1] => 3
001100 => [3,1,3] => 4
001101 => [3,1,2,1] => 1
001110 => [3,1,1,2] => 1
001111 => [3,1,1,1,1] => 1
010000 => [2,5] => 9
010001 => [2,4,1] => 5
010010 => [2,3,2] => 7
010011 => [2,3,1,1] => 2
010100 => [2,2,3] => 5
010101 => [2,2,2,1] => 5
010110 => [2,2,1,2] => 2
010111 => [2,2,1,1,1] => 2
011000 => [2,1,4] => 9
011001 => [2,1,3,1] => 3
011010 => [2,1,2,2] => 2
011011 => [2,1,2,1,1] => 1
011100 => [2,1,1,3] => 4
011101 => [2,1,1,2,1] => 1
011110 => [2,1,1,1,2] => 1
011111 => [2,1,1,1,1,1] => 1
100000 => [1,6] => 5
100001 => [1,5,1] => 4
100010 => [1,4,2] => 9
100011 => [1,4,1,1] => 3
100100 => [1,3,3] => 12
100101 => [1,3,2,1] => 5
100110 => [1,3,1,2] => 2
>>> Load all 511 entries. <<<
100111 => [1,3,1,1,1] => 2
101000 => [1,2,4] => 7
101001 => [1,2,3,1] => 2
101010 => [1,2,2,2] => 5
101011 => [1,2,2,1,1] => 2
101100 => [1,2,1,3] => 3
101101 => [1,2,1,2,1] => 1
101110 => [1,2,1,1,2] => 1
101111 => [1,2,1,1,1,1] => 1
110000 => [1,1,5] => 10
110001 => [1,1,4,1] => 6
110010 => [1,1,3,2] => 7
110011 => [1,1,3,1,1] => 3
110100 => [1,1,2,3] => 2
110101 => [1,1,2,2,1] => 2
110110 => [1,1,2,1,2] => 1
110111 => [1,1,2,1,1,1] => 1
111000 => [1,1,1,4] => 10
111001 => [1,1,1,3,1] => 4
111010 => [1,1,1,2,2] => 2
111011 => [1,1,1,2,1,1] => 1
111100 => [1,1,1,1,3] => 5
111101 => [1,1,1,1,2,1] => 1
111110 => [1,1,1,1,1,2] => 1
111111 => [1,1,1,1,1,1,1] => 1
0000000 => [8] => 1
0000001 => [7,1] => 1
0000010 => [6,2] => 6
0000011 => [6,1,1] => 1
0000100 => [5,3] => 14
0000101 => [5,2,1] => 5
0000110 => [5,1,2] => 1
0000111 => [5,1,1,1] => 1
0001000 => [4,4] => 14
0001001 => [4,3,1] => 9
0001010 => [4,2,2] => 14
0001011 => [4,2,1,1] => 4
0001100 => [4,1,3] => 5
0001101 => [4,1,2,1] => 1
0001110 => [4,1,1,2] => 1
0001111 => [4,1,1,1,1] => 1
0010000 => [3,5] => 14
0010001 => [3,4,1] => 5
0010010 => [3,3,2] => 21
0010011 => [3,3,1,1] => 5
0010100 => [3,2,3] => 9
0010101 => [3,2,2,1] => 9
0010110 => [3,2,1,2] => 3
0010111 => [3,2,1,1,1] => 3
0011000 => [3,1,4] => 9
0011001 => [3,1,3,1] => 4
0011010 => [3,1,2,2] => 2
0011011 => [3,1,2,1,1] => 1
0011100 => [3,1,1,3] => 5
0011101 => [3,1,1,2,1] => 1
0011110 => [3,1,1,1,2] => 1
0011111 => [3,1,1,1,1,1] => 1
0100000 => [2,6] => 14
0100001 => [2,5,1] => 9
0100010 => [2,4,2] => 21
0100011 => [2,4,1,1] => 5
0100100 => [2,3,3] => 12
0100101 => [2,3,2,1] => 7
0100110 => [2,3,1,2] => 2
0100111 => [2,3,1,1,1] => 2
0101000 => [2,2,4] => 21
0101001 => [2,2,3,1] => 5
0101010 => [2,2,2,2] => 14
0101011 => [2,2,2,1,1] => 5
0101100 => [2,2,1,3] => 7
0101101 => [2,2,1,2,1] => 2
0101110 => [2,2,1,1,2] => 2
0101111 => [2,2,1,1,1,1] => 2
0110000 => [2,1,5] => 19
0110001 => [2,1,4,1] => 9
0110010 => [2,1,3,2] => 7
0110011 => [2,1,3,1,1] => 3
0110100 => [2,1,2,3] => 2
0110101 => [2,1,2,2,1] => 2
0110110 => [2,1,2,1,2] => 1
0110111 => [2,1,2,1,1,1] => 1
0111000 => [2,1,1,4] => 14
0111001 => [2,1,1,3,1] => 4
0111010 => [2,1,1,2,2] => 2
0111011 => [2,1,1,2,1,1] => 1
0111100 => [2,1,1,1,3] => 5
0111101 => [2,1,1,1,2,1] => 1
0111110 => [2,1,1,1,1,2] => 1
0111111 => [2,1,1,1,1,1,1] => 1
1000000 => [1,7] => 6
1000001 => [1,6,1] => 5
1000010 => [1,5,2] => 14
1000011 => [1,5,1,1] => 4
1000100 => [1,4,3] => 30
1000101 => [1,4,2,1] => 9
1000110 => [1,4,1,2] => 3
1000111 => [1,4,1,1,1] => 3
1001000 => [1,3,4] => 12
1001001 => [1,3,3,1] => 12
1001010 => [1,3,2,2] => 14
1001011 => [1,3,2,1,1] => 5
1001100 => [1,3,1,3] => 9
1001101 => [1,3,1,2,1] => 2
1001110 => [1,3,1,1,2] => 2
1001111 => [1,3,1,1,1,1] => 2
1010000 => [1,2,5] => 16
1010001 => [1,2,4,1] => 7
1010010 => [1,2,3,2] => 7
1010011 => [1,2,3,1,1] => 2
1010100 => [1,2,2,3] => 5
1010101 => [1,2,2,2,1] => 5
1010110 => [1,2,2,1,2] => 2
1010111 => [1,2,2,1,1,1] => 2
1011000 => [1,2,1,4] => 12
1011001 => [1,2,1,3,1] => 3
1011010 => [1,2,1,2,2] => 2
1011011 => [1,2,1,2,1,1] => 1
1011100 => [1,2,1,1,3] => 4
1011101 => [1,2,1,1,2,1] => 1
1011110 => [1,2,1,1,1,2] => 1
1011111 => [1,2,1,1,1,1,1] => 1
1100000 => [1,1,6] => 15
1100001 => [1,1,5,1] => 10
1100010 => [1,1,4,2] => 16
1100011 => [1,1,4,1,1] => 6
1100100 => [1,1,3,3] => 21
1100101 => [1,1,3,2,1] => 7
1100110 => [1,1,3,1,2] => 3
1100111 => [1,1,3,1,1,1] => 3
1101000 => [1,1,2,4] => 9
1101001 => [1,1,2,3,1] => 2
1101010 => [1,1,2,2,2] => 5
1101011 => [1,1,2,2,1,1] => 2
1101100 => [1,1,2,1,3] => 3
1101101 => [1,1,2,1,2,1] => 1
1101110 => [1,1,2,1,1,2] => 1
1101111 => [1,1,2,1,1,1,1] => 1
1110000 => [1,1,1,5] => 20
1110001 => [1,1,1,4,1] => 10
1110010 => [1,1,1,3,2] => 9
1110011 => [1,1,1,3,1,1] => 4
1110100 => [1,1,1,2,3] => 2
1110101 => [1,1,1,2,2,1] => 2
1110110 => [1,1,1,2,1,2] => 1
1110111 => [1,1,1,2,1,1,1] => 1
1111000 => [1,1,1,1,4] => 15
1111001 => [1,1,1,1,3,1] => 5
1111010 => [1,1,1,1,2,2] => 2
1111011 => [1,1,1,1,2,1,1] => 1
1111100 => [1,1,1,1,1,3] => 6
1111101 => [1,1,1,1,1,2,1] => 1
1111110 => [1,1,1,1,1,1,2] => 1
1111111 => [1,1,1,1,1,1,1,1] => 1
00000000 => [9] => 1
00000001 => [8,1] => 1
00000010 => [7,2] => 7
00000011 => [7,1,1] => 1
00000100 => [6,3] => 20
00000101 => [6,2,1] => 6
00000110 => [6,1,2] => 1
00000111 => [6,1,1,1] => 1
00001000 => [5,4] => 28
00001001 => [5,3,1] => 14
00001010 => [5,2,2] => 20
00001011 => [5,2,1,1] => 5
00001100 => [5,1,3] => 6
00001101 => [5,1,2,1] => 1
00001110 => [5,1,1,2] => 1
00001111 => [5,1,1,1,1] => 1
00010000 => [4,5] => 14
00010001 => [4,4,1] => 14
00010010 => [4,3,2] => 44
00010011 => [4,3,1,1] => 9
00010100 => [4,2,3] => 14
00010101 => [4,2,2,1] => 14
00010110 => [4,2,1,2] => 4
00010111 => [4,2,1,1,1] => 4
00011000 => [4,1,4] => 14
00011001 => [4,1,3,1] => 5
00011010 => [4,1,2,2] => 2
00011011 => [4,1,2,1,1] => 1
00011100 => [4,1,1,3] => 6
00011101 => [4,1,1,2,1] => 1
00011110 => [4,1,1,1,2] => 1
00011111 => [4,1,1,1,1,1] => 1
00100000 => [3,6] => 28
00100001 => [3,5,1] => 14
00100010 => [3,4,2] => 26
00100011 => [3,4,1,1] => 5
00100100 => [3,3,3] => 42
00100101 => [3,3,2,1] => 21
00100110 => [3,3,1,2] => 5
00100111 => [3,3,1,1,1] => 5
00101000 => [3,2,4] => 30
00101001 => [3,2,3,1] => 9
00101010 => [3,2,2,2] => 28
00101011 => [3,2,2,1,1] => 9
00101100 => [3,2,1,3] => 12
00101101 => [3,2,1,2,1] => 3
00101110 => [3,2,1,1,2] => 3
00101111 => [3,2,1,1,1,1] => 3
00110000 => [3,1,5] => 28
00110001 => [3,1,4,1] => 9
00110010 => [3,1,3,2] => 9
00110011 => [3,1,3,1,1] => 4
00110100 => [3,1,2,3] => 2
00110101 => [3,1,2,2,1] => 2
00110110 => [3,1,2,1,2] => 1
00110111 => [3,1,2,1,1,1] => 1
00111000 => [3,1,1,4] => 14
00111001 => [3,1,1,3,1] => 5
00111010 => [3,1,1,2,2] => 2
00111011 => [3,1,1,2,1,1] => 1
00111100 => [3,1,1,1,3] => 6
00111101 => [3,1,1,1,2,1] => 1
00111110 => [3,1,1,1,1,2] => 1
00111111 => [3,1,1,1,1,1,1] => 1
01000000 => [2,7] => 20
01000001 => [2,6,1] => 14
01000010 => [2,5,2] => 44
01000011 => [2,5,1,1] => 9
01000100 => [2,4,3] => 42
01000101 => [2,4,2,1] => 21
01000110 => [2,4,1,2] => 5
01000111 => [2,4,1,1,1] => 5
01001000 => [2,3,4] => 12
01001001 => [2,3,3,1] => 12
01001010 => [2,3,2,2] => 23
01001011 => [2,3,2,1,1] => 7
01001100 => [2,3,1,3] => 9
01001101 => [2,3,1,2,1] => 2
01001110 => [2,3,1,1,2] => 2
01001111 => [2,3,1,1,1,1] => 2
01010000 => [2,2,5] => 56
01010001 => [2,2,4,1] => 21
01010010 => [2,2,3,2] => 19
01010011 => [2,2,3,1,1] => 5
01010100 => [2,2,2,3] => 14
01010101 => [2,2,2,2,1] => 14
01010110 => [2,2,2,1,2] => 5
01010111 => [2,2,2,1,1,1] => 5
01011000 => [2,2,1,4] => 33
01011001 => [2,2,1,3,1] => 7
01011010 => [2,2,1,2,2] => 4
01011011 => [2,2,1,2,1,1] => 2
01011100 => [2,2,1,1,3] => 9
01011101 => [2,2,1,1,2,1] => 2
01011110 => [2,2,1,1,1,2] => 2
01011111 => [2,2,1,1,1,1,1] => 2
01100000 => [2,1,6] => 34
01100001 => [2,1,5,1] => 19
01100010 => [2,1,4,2] => 23
01100011 => [2,1,4,1,1] => 9
01100100 => [2,1,3,3] => 21
01100101 => [2,1,3,2,1] => 7
01100110 => [2,1,3,1,2] => 3
01100111 => [2,1,3,1,1,1] => 3
01101000 => [2,1,2,4] => 11
01101001 => [2,1,2,3,1] => 2
01101010 => [2,1,2,2,2] => 5
01101011 => [2,1,2,2,1,1] => 2
01101100 => [2,1,2,1,3] => 3
01101101 => [2,1,2,1,2,1] => 1
01101110 => [2,1,2,1,1,2] => 1
01101111 => [2,1,2,1,1,1,1] => 1
01110000 => [2,1,1,5] => 34
01110001 => [2,1,1,4,1] => 14
01110010 => [2,1,1,3,2] => 9
01110011 => [2,1,1,3,1,1] => 4
01110100 => [2,1,1,2,3] => 2
01110101 => [2,1,1,2,2,1] => 2
01110110 => [2,1,1,2,1,2] => 1
01110111 => [2,1,1,2,1,1,1] => 1
01111000 => [2,1,1,1,4] => 20
01111001 => [2,1,1,1,3,1] => 5
01111010 => [2,1,1,1,2,2] => 2
01111011 => [2,1,1,1,2,1,1] => 1
01111100 => [2,1,1,1,1,3] => 6
01111101 => [2,1,1,1,1,2,1] => 1
01111110 => [2,1,1,1,1,1,2] => 1
01111111 => [2,1,1,1,1,1,1,1] => 1
10000000 => [1,8] => 7
10000001 => [1,7,1] => 6
10000010 => [1,6,2] => 20
10000011 => [1,6,1,1] => 5
10000100 => [1,5,3] => 58
10000101 => [1,5,2,1] => 14
10000110 => [1,5,1,2] => 4
10000111 => [1,5,1,1,1] => 4
10001000 => [1,4,4] => 56
10001001 => [1,4,3,1] => 30
10001010 => [1,4,2,2] => 28
10001011 => [1,4,2,1,1] => 9
10001100 => [1,4,1,3] => 17
10001101 => [1,4,1,2,1] => 3
10001110 => [1,4,1,1,2] => 3
10001111 => [1,4,1,1,1,1] => 3
10010000 => [1,3,5] => 42
10010001 => [1,3,4,1] => 12
10010010 => [1,3,3,2] => 42
10010011 => [1,3,3,1,1] => 12
10010100 => [1,3,2,3] => 14
10010101 => [1,3,2,2,1] => 14
10010110 => [1,3,2,1,2] => 5
10010111 => [1,3,2,1,1,1] => 5
10011000 => [1,3,1,4] => 21
10011001 => [1,3,1,3,1] => 9
10011010 => [1,3,1,2,2] => 4
10011011 => [1,3,1,2,1,1] => 2
10011100 => [1,3,1,1,3] => 11
10011101 => [1,3,1,1,2,1] => 2
10011110 => [1,3,1,1,1,2] => 2
10011111 => [1,3,1,1,1,1,1] => 2
10100000 => [1,2,6] => 30
10100001 => [1,2,5,1] => 16
10100010 => [1,2,4,2] => 28
10100011 => [1,2,4,1,1] => 7
10100100 => [1,2,3,3] => 12
10100101 => [1,2,3,2,1] => 7
10100110 => [1,2,3,1,2] => 2
10100111 => [1,2,3,1,1,1] => 2
10101000 => [1,2,2,4] => 26
10101001 => [1,2,2,3,1] => 5
10101010 => [1,2,2,2,2] => 14
10101011 => [1,2,2,2,1,1] => 5
10101100 => [1,2,2,1,3] => 7
10101101 => [1,2,2,1,2,1] => 2
10101110 => [1,2,2,1,1,2] => 2
10101111 => [1,2,2,1,1,1,1] => 2
10110000 => [1,2,1,5] => 31
10110001 => [1,2,1,4,1] => 12
10110010 => [1,2,1,3,2] => 7
10110011 => [1,2,1,3,1,1] => 3
10110100 => [1,2,1,2,3] => 2
10110101 => [1,2,1,2,2,1] => 2
10110110 => [1,2,1,2,1,2] => 1
10110111 => [1,2,1,2,1,1,1] => 1
10111000 => [1,2,1,1,4] => 18
10111001 => [1,2,1,1,3,1] => 4
10111010 => [1,2,1,1,2,2] => 2
10111011 => [1,2,1,1,2,1,1] => 1
10111100 => [1,2,1,1,1,3] => 5
10111101 => [1,2,1,1,1,2,1] => 1
10111110 => [1,2,1,1,1,1,2] => 1
10111111 => [1,2,1,1,1,1,1,1] => 1
11000000 => [1,1,7] => 21
11000001 => [1,1,6,1] => 15
11000010 => [1,1,5,2] => 30
11000011 => [1,1,5,1,1] => 10
11000100 => [1,1,4,3] => 67
11000101 => [1,1,4,2,1] => 16
11000110 => [1,1,4,1,2] => 6
11000111 => [1,1,4,1,1,1] => 6
11001000 => [1,1,3,4] => 21
11001001 => [1,1,3,3,1] => 21
11001010 => [1,1,3,2,2] => 19
11001011 => [1,1,3,2,1,1] => 7
11001100 => [1,1,3,1,3] => 15
11001101 => [1,1,3,1,2,1] => 3
11001110 => [1,1,3,1,1,2] => 3
11001111 => [1,1,3,1,1,1,1] => 3
11010000 => [1,1,2,5] => 25
11010001 => [1,1,2,4,1] => 9
11010010 => [1,1,2,3,2] => 7
11010011 => [1,1,2,3,1,1] => 2
11010100 => [1,1,2,2,3] => 5
11010101 => [1,1,2,2,2,1] => 5
11010110 => [1,1,2,2,1,2] => 2
11010111 => [1,1,2,2,1,1,1] => 2
11011000 => [1,1,2,1,4] => 15
11011001 => [1,1,2,1,3,1] => 3
11011010 => [1,1,2,1,2,2] => 2
11011011 => [1,1,2,1,2,1,1] => 1
11011100 => [1,1,2,1,1,3] => 4
11011101 => [1,1,2,1,1,2,1] => 1
11011110 => [1,1,2,1,1,1,2] => 1
11011111 => [1,1,2,1,1,1,1,1] => 1
11100000 => [1,1,1,6] => 35
11100001 => [1,1,1,5,1] => 20
11100010 => [1,1,1,4,2] => 25
11100011 => [1,1,1,4,1,1] => 10
11100100 => [1,1,1,3,3] => 32
11100101 => [1,1,1,3,2,1] => 9
11100110 => [1,1,1,3,1,2] => 4
11100111 => [1,1,1,3,1,1,1] => 4
11101000 => [1,1,1,2,4] => 11
11101001 => [1,1,1,2,3,1] => 2
11101010 => [1,1,1,2,2,2] => 5
11101011 => [1,1,1,2,2,1,1] => 2
11101100 => [1,1,1,2,1,3] => 3
11101101 => [1,1,1,2,1,2,1] => 1
11101110 => [1,1,1,2,1,1,2] => 1
11101111 => [1,1,1,2,1,1,1,1] => 1
11110000 => [1,1,1,1,5] => 35
11110001 => [1,1,1,1,4,1] => 15
11110010 => [1,1,1,1,3,2] => 11
11110011 => [1,1,1,1,3,1,1] => 5
11110100 => [1,1,1,1,2,3] => 2
11110101 => [1,1,1,1,2,2,1] => 2
11110110 => [1,1,1,1,2,1,2] => 1
11110111 => [1,1,1,1,2,1,1,1] => 1
11111000 => [1,1,1,1,1,4] => 21
11111001 => [1,1,1,1,1,3,1] => 6
11111010 => [1,1,1,1,1,2,2] => 2
11111011 => [1,1,1,1,1,2,1,1] => 1
11111100 => [1,1,1,1,1,1,3] => 7
11111101 => [1,1,1,1,1,1,2,1] => 1
11111110 => [1,1,1,1,1,1,1,2] => 1
11111111 => [1,1,1,1,1,1,1,1,1] => 1
=> [1] => 1
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Description
The number of standard composition tableaux of the composition.
See [1, Def. 4.2.6].
Apparently, the total number of tableaux of given size is the number of involutions.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.