Identifier
-
Mp00097:
Binary words
—delta morphism⟶
Integer compositions
St000816: Integer compositions ⟶ ℤ
Values
0 => [1] => 1
1 => [1] => 1
00 => [2] => 1
01 => [1,1] => 1
10 => [1,1] => 1
11 => [2] => 1
000 => [3] => 1
001 => [2,1] => 1
010 => [1,1,1] => 1
011 => [1,2] => 1
100 => [1,2] => 1
101 => [1,1,1] => 1
110 => [2,1] => 1
111 => [3] => 1
0000 => [4] => 1
0001 => [3,1] => 1
0010 => [2,1,1] => 1
0011 => [2,2] => 2
0100 => [1,1,2] => 1
0101 => [1,1,1,1] => 1
0110 => [1,2,1] => 1
0111 => [1,3] => 2
1000 => [1,3] => 2
1001 => [1,2,1] => 1
1010 => [1,1,1,1] => 1
1011 => [1,1,2] => 1
1100 => [2,2] => 2
1101 => [2,1,1] => 1
1110 => [3,1] => 1
1111 => [4] => 1
00000 => [5] => 1
00001 => [4,1] => 1
00010 => [3,1,1] => 1
00011 => [3,2] => 3
00100 => [2,1,2] => 1
00101 => [2,1,1,1] => 1
00110 => [2,2,1] => 2
00111 => [2,3] => 2
01000 => [1,1,3] => 3
01001 => [1,1,2,1] => 1
01010 => [1,1,1,1,1] => 1
01011 => [1,1,1,2] => 1
01100 => [1,2,2] => 2
01101 => [1,2,1,1] => 1
01110 => [1,3,1] => 2
01111 => [1,4] => 3
10000 => [1,4] => 3
10001 => [1,3,1] => 2
10010 => [1,2,1,1] => 1
10011 => [1,2,2] => 2
10100 => [1,1,1,2] => 1
10101 => [1,1,1,1,1] => 1
10110 => [1,1,2,1] => 1
10111 => [1,1,3] => 3
11000 => [2,3] => 2
11001 => [2,2,1] => 2
11010 => [2,1,1,1] => 1
11011 => [2,1,2] => 1
11100 => [3,2] => 3
11101 => [3,1,1] => 1
11110 => [4,1] => 1
11111 => [5] => 1
000000 => [6] => 1
000001 => [5,1] => 1
000010 => [4,1,1] => 1
000011 => [4,2] => 4
000100 => [3,1,2] => 1
000101 => [3,1,1,1] => 1
000110 => [3,2,1] => 3
000111 => [3,3] => 5
001000 => [2,1,3] => 3
001001 => [2,1,2,1] => 1
001010 => [2,1,1,1,1] => 1
001011 => [2,1,1,2] => 1
001100 => [2,2,2] => 5
001101 => [2,2,1,1] => 2
001110 => [2,3,1] => 2
001111 => [2,4] => 5
010000 => [1,1,4] => 6
010001 => [1,1,3,1] => 3
010010 => [1,1,2,1,1] => 1
010011 => [1,1,2,2] => 2
010100 => [1,1,1,1,2] => 1
010101 => [1,1,1,1,1,1] => 1
010110 => [1,1,1,2,1] => 1
010111 => [1,1,1,3] => 4
011000 => [1,2,3] => 2
011001 => [1,2,2,1] => 2
011010 => [1,2,1,1,1] => 1
011011 => [1,2,1,2] => 1
011100 => [1,3,2] => 5
011101 => [1,3,1,1] => 2
011110 => [1,4,1] => 3
011111 => [1,5] => 4
100000 => [1,5] => 4
100001 => [1,4,1] => 3
100010 => [1,3,1,1] => 2
100011 => [1,3,2] => 5
100100 => [1,2,1,2] => 1
100101 => [1,2,1,1,1] => 1
100110 => [1,2,2,1] => 2
>>> Load all 1022 entries. <<<100111 => [1,2,3] => 2
101000 => [1,1,1,3] => 4
101001 => [1,1,1,2,1] => 1
101010 => [1,1,1,1,1,1] => 1
101011 => [1,1,1,1,2] => 1
101100 => [1,1,2,2] => 2
101101 => [1,1,2,1,1] => 1
101110 => [1,1,3,1] => 3
101111 => [1,1,4] => 6
110000 => [2,4] => 5
110001 => [2,3,1] => 2
110010 => [2,2,1,1] => 2
110011 => [2,2,2] => 5
110100 => [2,1,1,2] => 1
110101 => [2,1,1,1,1] => 1
110110 => [2,1,2,1] => 1
110111 => [2,1,3] => 3
111000 => [3,3] => 5
111001 => [3,2,1] => 3
111010 => [3,1,1,1] => 1
111011 => [3,1,2] => 1
111100 => [4,2] => 4
111101 => [4,1,1] => 1
111110 => [5,1] => 1
111111 => [6] => 1
0000000 => [7] => 1
0000001 => [6,1] => 1
0000010 => [5,1,1] => 1
0000011 => [5,2] => 5
0000100 => [4,1,2] => 1
0000101 => [4,1,1,1] => 1
0000110 => [4,2,1] => 4
0000111 => [4,3] => 9
0001000 => [3,1,3] => 4
0001001 => [3,1,2,1] => 1
0001010 => [3,1,1,1,1] => 1
0001011 => [3,1,1,2] => 1
0001100 => [3,2,2] => 9
0001101 => [3,2,1,1] => 3
0001110 => [3,3,1] => 5
0001111 => [3,4] => 5
0010000 => [2,1,4] => 9
0010001 => [2,1,3,1] => 3
0010010 => [2,1,2,1,1] => 1
0010011 => [2,1,2,2] => 2
0010100 => [2,1,1,1,2] => 1
0010101 => [2,1,1,1,1,1] => 1
0010110 => [2,1,1,2,1] => 1
0010111 => [2,1,1,3] => 4
0011000 => [2,2,3] => 5
0011001 => [2,2,2,1] => 5
0011010 => [2,2,1,1,1] => 2
0011011 => [2,2,1,2] => 2
0011100 => [2,3,2] => 7
0011101 => [2,3,1,1] => 2
0011110 => [2,4,1] => 5
0011111 => [2,5] => 9
0100000 => [1,1,5] => 10
0100001 => [1,1,4,1] => 6
0100010 => [1,1,3,1,1] => 3
0100011 => [1,1,3,2] => 7
0100100 => [1,1,2,1,2] => 1
0100101 => [1,1,2,1,1,1] => 1
0100110 => [1,1,2,2,1] => 2
0100111 => [1,1,2,3] => 2
0101000 => [1,1,1,1,3] => 5
0101001 => [1,1,1,1,2,1] => 1
0101010 => [1,1,1,1,1,1,1] => 1
0101011 => [1,1,1,1,1,2] => 1
0101100 => [1,1,1,2,2] => 2
0101101 => [1,1,1,2,1,1] => 1
0101110 => [1,1,1,3,1] => 4
0101111 => [1,1,1,4] => 10
0110000 => [1,2,4] => 7
0110001 => [1,2,3,1] => 2
0110010 => [1,2,2,1,1] => 2
0110011 => [1,2,2,2] => 5
0110100 => [1,2,1,1,2] => 1
0110101 => [1,2,1,1,1,1] => 1
0110110 => [1,2,1,2,1] => 1
0110111 => [1,2,1,3] => 3
0111000 => [1,3,3] => 12
0111001 => [1,3,2,1] => 5
0111010 => [1,3,1,1,1] => 2
0111011 => [1,3,1,2] => 2
0111100 => [1,4,2] => 9
0111101 => [1,4,1,1] => 3
0111110 => [1,5,1] => 4
0111111 => [1,6] => 5
1000000 => [1,6] => 5
1000001 => [1,5,1] => 4
1000010 => [1,4,1,1] => 3
1000011 => [1,4,2] => 9
1000100 => [1,3,1,2] => 2
1000101 => [1,3,1,1,1] => 2
1000110 => [1,3,2,1] => 5
1000111 => [1,3,3] => 12
1001000 => [1,2,1,3] => 3
1001001 => [1,2,1,2,1] => 1
1001010 => [1,2,1,1,1,1] => 1
1001011 => [1,2,1,1,2] => 1
1001100 => [1,2,2,2] => 5
1001101 => [1,2,2,1,1] => 2
1001110 => [1,2,3,1] => 2
1001111 => [1,2,4] => 7
1010000 => [1,1,1,4] => 10
1010001 => [1,1,1,3,1] => 4
1010010 => [1,1,1,2,1,1] => 1
1010011 => [1,1,1,2,2] => 2
1010100 => [1,1,1,1,1,2] => 1
1010101 => [1,1,1,1,1,1,1] => 1
1010110 => [1,1,1,1,2,1] => 1
1010111 => [1,1,1,1,3] => 5
1011000 => [1,1,2,3] => 2
1011001 => [1,1,2,2,1] => 2
1011010 => [1,1,2,1,1,1] => 1
1011011 => [1,1,2,1,2] => 1
1011100 => [1,1,3,2] => 7
1011101 => [1,1,3,1,1] => 3
1011110 => [1,1,4,1] => 6
1011111 => [1,1,5] => 10
1100000 => [2,5] => 9
1100001 => [2,4,1] => 5
1100010 => [2,3,1,1] => 2
1100011 => [2,3,2] => 7
1100100 => [2,2,1,2] => 2
1100101 => [2,2,1,1,1] => 2
1100110 => [2,2,2,1] => 5
1100111 => [2,2,3] => 5
1101000 => [2,1,1,3] => 4
1101001 => [2,1,1,2,1] => 1
1101010 => [2,1,1,1,1,1] => 1
1101011 => [2,1,1,1,2] => 1
1101100 => [2,1,2,2] => 2
1101101 => [2,1,2,1,1] => 1
1101110 => [2,1,3,1] => 3
1101111 => [2,1,4] => 9
1110000 => [3,4] => 5
1110001 => [3,3,1] => 5
1110010 => [3,2,1,1] => 3
1110011 => [3,2,2] => 9
1110100 => [3,1,1,2] => 1
1110101 => [3,1,1,1,1] => 1
1110110 => [3,1,2,1] => 1
1110111 => [3,1,3] => 4
1111000 => [4,3] => 9
1111001 => [4,2,1] => 4
1111010 => [4,1,1,1] => 1
1111011 => [4,1,2] => 1
1111100 => [5,2] => 5
1111101 => [5,1,1] => 1
1111110 => [6,1] => 1
1111111 => [7] => 1
00000000 => [8] => 1
00000001 => [7,1] => 1
00000010 => [6,1,1] => 1
00000011 => [6,2] => 6
00000100 => [5,1,2] => 1
00000101 => [5,1,1,1] => 1
00000110 => [5,2,1] => 5
00000111 => [5,3] => 14
00001000 => [4,1,3] => 5
00001001 => [4,1,2,1] => 1
00001010 => [4,1,1,1,1] => 1
00001011 => [4,1,1,2] => 1
00001100 => [4,2,2] => 14
00001101 => [4,2,1,1] => 4
00001110 => [4,3,1] => 9
00001111 => [4,4] => 14
00010000 => [3,1,4] => 9
00010001 => [3,1,3,1] => 4
00010010 => [3,1,2,1,1] => 1
00010011 => [3,1,2,2] => 2
00010100 => [3,1,1,1,2] => 1
00010101 => [3,1,1,1,1,1] => 1
00010110 => [3,1,1,2,1] => 1
00010111 => [3,1,1,3] => 5
00011000 => [3,2,3] => 9
00011001 => [3,2,2,1] => 9
00011010 => [3,2,1,1,1] => 3
00011011 => [3,2,1,2] => 3
00011100 => [3,3,2] => 21
00011101 => [3,3,1,1] => 5
00011110 => [3,4,1] => 5
00011111 => [3,5] => 14
00100000 => [2,1,5] => 19
00100001 => [2,1,4,1] => 9
00100010 => [2,1,3,1,1] => 3
00100011 => [2,1,3,2] => 7
00100100 => [2,1,2,1,2] => 1
00100101 => [2,1,2,1,1,1] => 1
00100110 => [2,1,2,2,1] => 2
00100111 => [2,1,2,3] => 2
00101000 => [2,1,1,1,3] => 5
00101001 => [2,1,1,1,2,1] => 1
00101010 => [2,1,1,1,1,1,1] => 1
00101011 => [2,1,1,1,1,2] => 1
00101100 => [2,1,1,2,2] => 2
00101101 => [2,1,1,2,1,1] => 1
00101110 => [2,1,1,3,1] => 4
00101111 => [2,1,1,4] => 14
00110000 => [2,2,4] => 21
00110001 => [2,2,3,1] => 5
00110010 => [2,2,2,1,1] => 5
00110011 => [2,2,2,2] => 14
00110100 => [2,2,1,1,2] => 2
00110101 => [2,2,1,1,1,1] => 2
00110110 => [2,2,1,2,1] => 2
00110111 => [2,2,1,3] => 7
00111000 => [2,3,3] => 12
00111001 => [2,3,2,1] => 7
00111010 => [2,3,1,1,1] => 2
00111011 => [2,3,1,2] => 2
00111100 => [2,4,2] => 21
00111101 => [2,4,1,1] => 5
00111110 => [2,5,1] => 9
00111111 => [2,6] => 14
01000000 => [1,1,6] => 15
01000001 => [1,1,5,1] => 10
01000010 => [1,1,4,1,1] => 6
01000011 => [1,1,4,2] => 16
01000100 => [1,1,3,1,2] => 3
01000101 => [1,1,3,1,1,1] => 3
01000110 => [1,1,3,2,1] => 7
01000111 => [1,1,3,3] => 21
01001000 => [1,1,2,1,3] => 3
01001001 => [1,1,2,1,2,1] => 1
01001010 => [1,1,2,1,1,1,1] => 1
01001011 => [1,1,2,1,1,2] => 1
01001100 => [1,1,2,2,2] => 5
01001101 => [1,1,2,2,1,1] => 2
01001110 => [1,1,2,3,1] => 2
01001111 => [1,1,2,4] => 9
01010000 => [1,1,1,1,4] => 15
01010001 => [1,1,1,1,3,1] => 5
01010010 => [1,1,1,1,2,1,1] => 1
01010011 => [1,1,1,1,2,2] => 2
01010100 => [1,1,1,1,1,1,2] => 1
01010101 => [1,1,1,1,1,1,1,1] => 1
01010110 => [1,1,1,1,1,2,1] => 1
01010111 => [1,1,1,1,1,3] => 6
01011000 => [1,1,1,2,3] => 2
01011001 => [1,1,1,2,2,1] => 2
01011010 => [1,1,1,2,1,1,1] => 1
01011011 => [1,1,1,2,1,2] => 1
01011100 => [1,1,1,3,2] => 9
01011101 => [1,1,1,3,1,1] => 4
01011110 => [1,1,1,4,1] => 10
01011111 => [1,1,1,5] => 20
01100000 => [1,2,5] => 16
01100001 => [1,2,4,1] => 7
01100010 => [1,2,3,1,1] => 2
01100011 => [1,2,3,2] => 7
01100100 => [1,2,2,1,2] => 2
01100101 => [1,2,2,1,1,1] => 2
01100110 => [1,2,2,2,1] => 5
01100111 => [1,2,2,3] => 5
01101000 => [1,2,1,1,3] => 4
01101001 => [1,2,1,1,2,1] => 1
01101010 => [1,2,1,1,1,1,1] => 1
01101011 => [1,2,1,1,1,2] => 1
01101100 => [1,2,1,2,2] => 2
01101101 => [1,2,1,2,1,1] => 1
01101110 => [1,2,1,3,1] => 3
01101111 => [1,2,1,4] => 12
01110000 => [1,3,4] => 12
01110001 => [1,3,3,1] => 12
01110010 => [1,3,2,1,1] => 5
01110011 => [1,3,2,2] => 14
01110100 => [1,3,1,1,2] => 2
01110101 => [1,3,1,1,1,1] => 2
01110110 => [1,3,1,2,1] => 2
01110111 => [1,3,1,3] => 9
01111000 => [1,4,3] => 30
01111001 => [1,4,2,1] => 9
01111010 => [1,4,1,1,1] => 3
01111011 => [1,4,1,2] => 3
01111100 => [1,5,2] => 14
01111101 => [1,5,1,1] => 4
01111110 => [1,6,1] => 5
01111111 => [1,7] => 6
10000000 => [1,7] => 6
10000001 => [1,6,1] => 5
10000010 => [1,5,1,1] => 4
10000011 => [1,5,2] => 14
10000100 => [1,4,1,2] => 3
10000101 => [1,4,1,1,1] => 3
10000110 => [1,4,2,1] => 9
10000111 => [1,4,3] => 30
10001000 => [1,3,1,3] => 9
10001001 => [1,3,1,2,1] => 2
10001010 => [1,3,1,1,1,1] => 2
10001011 => [1,3,1,1,2] => 2
10001100 => [1,3,2,2] => 14
10001101 => [1,3,2,1,1] => 5
10001110 => [1,3,3,1] => 12
10001111 => [1,3,4] => 12
10010000 => [1,2,1,4] => 12
10010001 => [1,2,1,3,1] => 3
10010010 => [1,2,1,2,1,1] => 1
10010011 => [1,2,1,2,2] => 2
10010100 => [1,2,1,1,1,2] => 1
10010101 => [1,2,1,1,1,1,1] => 1
10010110 => [1,2,1,1,2,1] => 1
10010111 => [1,2,1,1,3] => 4
10011000 => [1,2,2,3] => 5
10011001 => [1,2,2,2,1] => 5
10011010 => [1,2,2,1,1,1] => 2
10011011 => [1,2,2,1,2] => 2
10011100 => [1,2,3,2] => 7
10011101 => [1,2,3,1,1] => 2
10011110 => [1,2,4,1] => 7
10011111 => [1,2,5] => 16
10100000 => [1,1,1,5] => 20
10100001 => [1,1,1,4,1] => 10
10100010 => [1,1,1,3,1,1] => 4
10100011 => [1,1,1,3,2] => 9
10100100 => [1,1,1,2,1,2] => 1
10100101 => [1,1,1,2,1,1,1] => 1
10100110 => [1,1,1,2,2,1] => 2
10100111 => [1,1,1,2,3] => 2
10101000 => [1,1,1,1,1,3] => 6
10101001 => [1,1,1,1,1,2,1] => 1
10101010 => [1,1,1,1,1,1,1,1] => 1
10101011 => [1,1,1,1,1,1,2] => 1
10101100 => [1,1,1,1,2,2] => 2
10101101 => [1,1,1,1,2,1,1] => 1
10101110 => [1,1,1,1,3,1] => 5
10101111 => [1,1,1,1,4] => 15
10110000 => [1,1,2,4] => 9
10110001 => [1,1,2,3,1] => 2
10110010 => [1,1,2,2,1,1] => 2
10110011 => [1,1,2,2,2] => 5
10110100 => [1,1,2,1,1,2] => 1
10110101 => [1,1,2,1,1,1,1] => 1
10110110 => [1,1,2,1,2,1] => 1
10110111 => [1,1,2,1,3] => 3
10111000 => [1,1,3,3] => 21
10111001 => [1,1,3,2,1] => 7
10111010 => [1,1,3,1,1,1] => 3
10111011 => [1,1,3,1,2] => 3
10111100 => [1,1,4,2] => 16
10111101 => [1,1,4,1,1] => 6
10111110 => [1,1,5,1] => 10
10111111 => [1,1,6] => 15
11000000 => [2,6] => 14
11000001 => [2,5,1] => 9
11000010 => [2,4,1,1] => 5
11000011 => [2,4,2] => 21
11000100 => [2,3,1,2] => 2
11000101 => [2,3,1,1,1] => 2
11000110 => [2,3,2,1] => 7
11000111 => [2,3,3] => 12
11001000 => [2,2,1,3] => 7
11001001 => [2,2,1,2,1] => 2
11001010 => [2,2,1,1,1,1] => 2
11001011 => [2,2,1,1,2] => 2
11001100 => [2,2,2,2] => 14
11001101 => [2,2,2,1,1] => 5
11001110 => [2,2,3,1] => 5
11001111 => [2,2,4] => 21
11010000 => [2,1,1,4] => 14
11010001 => [2,1,1,3,1] => 4
11010010 => [2,1,1,2,1,1] => 1
11010011 => [2,1,1,2,2] => 2
11010100 => [2,1,1,1,1,2] => 1
11010101 => [2,1,1,1,1,1,1] => 1
11010110 => [2,1,1,1,2,1] => 1
11010111 => [2,1,1,1,3] => 5
11011000 => [2,1,2,3] => 2
11011001 => [2,1,2,2,1] => 2
11011010 => [2,1,2,1,1,1] => 1
11011011 => [2,1,2,1,2] => 1
11011100 => [2,1,3,2] => 7
11011101 => [2,1,3,1,1] => 3
11011110 => [2,1,4,1] => 9
11011111 => [2,1,5] => 19
11100000 => [3,5] => 14
11100001 => [3,4,1] => 5
11100010 => [3,3,1,1] => 5
11100011 => [3,3,2] => 21
11100100 => [3,2,1,2] => 3
11100101 => [3,2,1,1,1] => 3
11100110 => [3,2,2,1] => 9
11100111 => [3,2,3] => 9
11101000 => [3,1,1,3] => 5
11101001 => [3,1,1,2,1] => 1
11101010 => [3,1,1,1,1,1] => 1
11101011 => [3,1,1,1,2] => 1
11101100 => [3,1,2,2] => 2
11101101 => [3,1,2,1,1] => 1
11101110 => [3,1,3,1] => 4
11101111 => [3,1,4] => 9
11110000 => [4,4] => 14
11110001 => [4,3,1] => 9
11110010 => [4,2,1,1] => 4
11110011 => [4,2,2] => 14
11110100 => [4,1,1,2] => 1
11110101 => [4,1,1,1,1] => 1
11110110 => [4,1,2,1] => 1
11110111 => [4,1,3] => 5
11111000 => [5,3] => 14
11111001 => [5,2,1] => 5
11111010 => [5,1,1,1] => 1
11111011 => [5,1,2] => 1
11111100 => [6,2] => 6
11111101 => [6,1,1] => 1
11111110 => [7,1] => 1
11111111 => [8] => 1
000000000 => [9] => 1
000000001 => [8,1] => 1
000000010 => [7,1,1] => 1
000000011 => [7,2] => 7
000000100 => [6,1,2] => 1
000000101 => [6,1,1,1] => 1
000000110 => [6,2,1] => 6
000000111 => [6,3] => 20
000001000 => [5,1,3] => 6
000001001 => [5,1,2,1] => 1
000001010 => [5,1,1,1,1] => 1
000001011 => [5,1,1,2] => 1
000001100 => [5,2,2] => 20
000001101 => [5,2,1,1] => 5
000001110 => [5,3,1] => 14
000001111 => [5,4] => 28
000010000 => [4,1,4] => 14
000010001 => [4,1,3,1] => 5
000010010 => [4,1,2,1,1] => 1
000010011 => [4,1,2,2] => 2
000010100 => [4,1,1,1,2] => 1
000010101 => [4,1,1,1,1,1] => 1
000010110 => [4,1,1,2,1] => 1
000010111 => [4,1,1,3] => 6
000011000 => [4,2,3] => 14
000011001 => [4,2,2,1] => 14
000011010 => [4,2,1,1,1] => 4
000011011 => [4,2,1,2] => 4
000011100 => [4,3,2] => 44
000011101 => [4,3,1,1] => 9
000011110 => [4,4,1] => 14
000011111 => [4,5] => 14
000100000 => [3,1,5] => 28
000100001 => [3,1,4,1] => 9
000100010 => [3,1,3,1,1] => 4
000100011 => [3,1,3,2] => 9
000100100 => [3,1,2,1,2] => 1
000100101 => [3,1,2,1,1,1] => 1
000100110 => [3,1,2,2,1] => 2
000100111 => [3,1,2,3] => 2
000101000 => [3,1,1,1,3] => 6
000101001 => [3,1,1,1,2,1] => 1
000101010 => [3,1,1,1,1,1,1] => 1
000101011 => [3,1,1,1,1,2] => 1
000101100 => [3,1,1,2,2] => 2
000101101 => [3,1,1,2,1,1] => 1
000101110 => [3,1,1,3,1] => 5
000101111 => [3,1,1,4] => 14
000110000 => [3,2,4] => 30
000110001 => [3,2,3,1] => 9
000110010 => [3,2,2,1,1] => 9
000110011 => [3,2,2,2] => 28
000110100 => [3,2,1,1,2] => 3
000110101 => [3,2,1,1,1,1] => 3
000110110 => [3,2,1,2,1] => 3
000110111 => [3,2,1,3] => 12
000111000 => [3,3,3] => 42
000111001 => [3,3,2,1] => 21
000111010 => [3,3,1,1,1] => 5
000111011 => [3,3,1,2] => 5
000111100 => [3,4,2] => 26
000111101 => [3,4,1,1] => 5
000111110 => [3,5,1] => 14
000111111 => [3,6] => 28
001000000 => [2,1,6] => 34
001000001 => [2,1,5,1] => 19
001000010 => [2,1,4,1,1] => 9
001000011 => [2,1,4,2] => 23
001000100 => [2,1,3,1,2] => 3
001000101 => [2,1,3,1,1,1] => 3
001000110 => [2,1,3,2,1] => 7
001000111 => [2,1,3,3] => 21
001001000 => [2,1,2,1,3] => 3
001001001 => [2,1,2,1,2,1] => 1
001001010 => [2,1,2,1,1,1,1] => 1
001001011 => [2,1,2,1,1,2] => 1
001001100 => [2,1,2,2,2] => 5
001001101 => [2,1,2,2,1,1] => 2
001001110 => [2,1,2,3,1] => 2
001001111 => [2,1,2,4] => 11
001010000 => [2,1,1,1,4] => 20
001010001 => [2,1,1,1,3,1] => 5
001010010 => [2,1,1,1,2,1,1] => 1
001010011 => [2,1,1,1,2,2] => 2
001010100 => [2,1,1,1,1,1,2] => 1
001010101 => [2,1,1,1,1,1,1,1] => 1
001010110 => [2,1,1,1,1,2,1] => 1
001010111 => [2,1,1,1,1,3] => 6
001011000 => [2,1,1,2,3] => 2
001011001 => [2,1,1,2,2,1] => 2
001011010 => [2,1,1,2,1,1,1] => 1
001011011 => [2,1,1,2,1,2] => 1
001011100 => [2,1,1,3,2] => 9
001011101 => [2,1,1,3,1,1] => 4
001011110 => [2,1,1,4,1] => 14
001011111 => [2,1,1,5] => 34
001100000 => [2,2,5] => 56
001100001 => [2,2,4,1] => 21
001100010 => [2,2,3,1,1] => 5
001100011 => [2,2,3,2] => 19
001100100 => [2,2,2,1,2] => 5
001100101 => [2,2,2,1,1,1] => 5
001100110 => [2,2,2,2,1] => 14
001100111 => [2,2,2,3] => 14
001101000 => [2,2,1,1,3] => 9
001101001 => [2,2,1,1,2,1] => 2
001101010 => [2,2,1,1,1,1,1] => 2
001101011 => [2,2,1,1,1,2] => 2
001101100 => [2,2,1,2,2] => 4
001101101 => [2,2,1,2,1,1] => 2
001101110 => [2,2,1,3,1] => 7
001101111 => [2,2,1,4] => 33
001110000 => [2,3,4] => 12
001110001 => [2,3,3,1] => 12
001110010 => [2,3,2,1,1] => 7
001110011 => [2,3,2,2] => 23
001110100 => [2,3,1,1,2] => 2
001110101 => [2,3,1,1,1,1] => 2
001110110 => [2,3,1,2,1] => 2
001110111 => [2,3,1,3] => 9
001111000 => [2,4,3] => 42
001111001 => [2,4,2,1] => 21
001111010 => [2,4,1,1,1] => 5
001111011 => [2,4,1,2] => 5
001111100 => [2,5,2] => 44
001111101 => [2,5,1,1] => 9
001111110 => [2,6,1] => 14
001111111 => [2,7] => 20
010000000 => [1,1,7] => 21
010000001 => [1,1,6,1] => 15
010000010 => [1,1,5,1,1] => 10
010000011 => [1,1,5,2] => 30
010000100 => [1,1,4,1,2] => 6
010000101 => [1,1,4,1,1,1] => 6
010000110 => [1,1,4,2,1] => 16
010000111 => [1,1,4,3] => 67
010001000 => [1,1,3,1,3] => 15
010001001 => [1,1,3,1,2,1] => 3
010001010 => [1,1,3,1,1,1,1] => 3
010001011 => [1,1,3,1,1,2] => 3
010001100 => [1,1,3,2,2] => 19
010001101 => [1,1,3,2,1,1] => 7
010001110 => [1,1,3,3,1] => 21
010001111 => [1,1,3,4] => 21
010010000 => [1,1,2,1,4] => 15
010010001 => [1,1,2,1,3,1] => 3
010010010 => [1,1,2,1,2,1,1] => 1
010010011 => [1,1,2,1,2,2] => 2
010010100 => [1,1,2,1,1,1,2] => 1
010010101 => [1,1,2,1,1,1,1,1] => 1
010010110 => [1,1,2,1,1,2,1] => 1
010010111 => [1,1,2,1,1,3] => 4
010011000 => [1,1,2,2,3] => 5
010011001 => [1,1,2,2,2,1] => 5
010011010 => [1,1,2,2,1,1,1] => 2
010011011 => [1,1,2,2,1,2] => 2
010011100 => [1,1,2,3,2] => 7
010011101 => [1,1,2,3,1,1] => 2
010011110 => [1,1,2,4,1] => 9
010011111 => [1,1,2,5] => 25
010100000 => [1,1,1,1,5] => 35
010100001 => [1,1,1,1,4,1] => 15
010100010 => [1,1,1,1,3,1,1] => 5
010100011 => [1,1,1,1,3,2] => 11
010100100 => [1,1,1,1,2,1,2] => 1
010100101 => [1,1,1,1,2,1,1,1] => 1
010100110 => [1,1,1,1,2,2,1] => 2
010100111 => [1,1,1,1,2,3] => 2
010101000 => [1,1,1,1,1,1,3] => 7
010101001 => [1,1,1,1,1,1,2,1] => 1
010101010 => [1,1,1,1,1,1,1,1,1] => 1
010101011 => [1,1,1,1,1,1,1,2] => 1
010101100 => [1,1,1,1,1,2,2] => 2
010101101 => [1,1,1,1,1,2,1,1] => 1
010101110 => [1,1,1,1,1,3,1] => 6
010101111 => [1,1,1,1,1,4] => 21
010110000 => [1,1,1,2,4] => 11
010110001 => [1,1,1,2,3,1] => 2
010110010 => [1,1,1,2,2,1,1] => 2
010110011 => [1,1,1,2,2,2] => 5
010110100 => [1,1,1,2,1,1,2] => 1
010110101 => [1,1,1,2,1,1,1,1] => 1
010110110 => [1,1,1,2,1,2,1] => 1
010110111 => [1,1,1,2,1,3] => 3
010111000 => [1,1,1,3,3] => 32
010111001 => [1,1,1,3,2,1] => 9
010111010 => [1,1,1,3,1,1,1] => 4
010111011 => [1,1,1,3,1,2] => 4
010111100 => [1,1,1,4,2] => 25
010111101 => [1,1,1,4,1,1] => 10
010111110 => [1,1,1,5,1] => 20
010111111 => [1,1,1,6] => 35
011000000 => [1,2,6] => 30
011000001 => [1,2,5,1] => 16
011000010 => [1,2,4,1,1] => 7
011000011 => [1,2,4,2] => 28
011000100 => [1,2,3,1,2] => 2
011000101 => [1,2,3,1,1,1] => 2
011000110 => [1,2,3,2,1] => 7
011000111 => [1,2,3,3] => 12
011001000 => [1,2,2,1,3] => 7
011001001 => [1,2,2,1,2,1] => 2
011001010 => [1,2,2,1,1,1,1] => 2
011001011 => [1,2,2,1,1,2] => 2
011001100 => [1,2,2,2,2] => 14
011001101 => [1,2,2,2,1,1] => 5
011001110 => [1,2,2,3,1] => 5
011001111 => [1,2,2,4] => 26
011010000 => [1,2,1,1,4] => 18
011010001 => [1,2,1,1,3,1] => 4
011010010 => [1,2,1,1,2,1,1] => 1
011010011 => [1,2,1,1,2,2] => 2
011010100 => [1,2,1,1,1,1,2] => 1
011010101 => [1,2,1,1,1,1,1,1] => 1
011010110 => [1,2,1,1,1,2,1] => 1
011010111 => [1,2,1,1,1,3] => 5
011011000 => [1,2,1,2,3] => 2
011011001 => [1,2,1,2,2,1] => 2
011011010 => [1,2,1,2,1,1,1] => 1
011011011 => [1,2,1,2,1,2] => 1
011011100 => [1,2,1,3,2] => 7
011011101 => [1,2,1,3,1,1] => 3
011011110 => [1,2,1,4,1] => 12
011011111 => [1,2,1,5] => 31
011100000 => [1,3,5] => 42
011100001 => [1,3,4,1] => 12
011100010 => [1,3,3,1,1] => 12
011100011 => [1,3,3,2] => 42
011100100 => [1,3,2,1,2] => 5
011100101 => [1,3,2,1,1,1] => 5
011100110 => [1,3,2,2,1] => 14
011100111 => [1,3,2,3] => 14
011101000 => [1,3,1,1,3] => 11
011101001 => [1,3,1,1,2,1] => 2
011101010 => [1,3,1,1,1,1,1] => 2
011101011 => [1,3,1,1,1,2] => 2
011101100 => [1,3,1,2,2] => 4
011101101 => [1,3,1,2,1,1] => 2
011101110 => [1,3,1,3,1] => 9
011101111 => [1,3,1,4] => 21
011110000 => [1,4,4] => 56
011110001 => [1,4,3,1] => 30
011110010 => [1,4,2,1,1] => 9
011110011 => [1,4,2,2] => 28
011110100 => [1,4,1,1,2] => 3
011110101 => [1,4,1,1,1,1] => 3
011110110 => [1,4,1,2,1] => 3
011110111 => [1,4,1,3] => 17
011111000 => [1,5,3] => 58
011111001 => [1,5,2,1] => 14
011111010 => [1,5,1,1,1] => 4
011111011 => [1,5,1,2] => 4
011111100 => [1,6,2] => 20
011111101 => [1,6,1,1] => 5
011111110 => [1,7,1] => 6
011111111 => [1,8] => 7
100000000 => [1,8] => 7
100000001 => [1,7,1] => 6
100000010 => [1,6,1,1] => 5
100000011 => [1,6,2] => 20
100000100 => [1,5,1,2] => 4
100000101 => [1,5,1,1,1] => 4
100000110 => [1,5,2,1] => 14
100000111 => [1,5,3] => 58
100001000 => [1,4,1,3] => 17
100001001 => [1,4,1,2,1] => 3
100001010 => [1,4,1,1,1,1] => 3
100001011 => [1,4,1,1,2] => 3
100001100 => [1,4,2,2] => 28
100001101 => [1,4,2,1,1] => 9
100001110 => [1,4,3,1] => 30
100001111 => [1,4,4] => 56
100010000 => [1,3,1,4] => 21
100010001 => [1,3,1,3,1] => 9
100010010 => [1,3,1,2,1,1] => 2
100010011 => [1,3,1,2,2] => 4
100010100 => [1,3,1,1,1,2] => 2
100010101 => [1,3,1,1,1,1,1] => 2
100010110 => [1,3,1,1,2,1] => 2
100010111 => [1,3,1,1,3] => 11
100011000 => [1,3,2,3] => 14
100011001 => [1,3,2,2,1] => 14
100011010 => [1,3,2,1,1,1] => 5
100011011 => [1,3,2,1,2] => 5
100011100 => [1,3,3,2] => 42
100011101 => [1,3,3,1,1] => 12
100011110 => [1,3,4,1] => 12
100011111 => [1,3,5] => 42
100100000 => [1,2,1,5] => 31
100100001 => [1,2,1,4,1] => 12
100100010 => [1,2,1,3,1,1] => 3
100100011 => [1,2,1,3,2] => 7
100100100 => [1,2,1,2,1,2] => 1
100100101 => [1,2,1,2,1,1,1] => 1
100100110 => [1,2,1,2,2,1] => 2
100100111 => [1,2,1,2,3] => 2
100101000 => [1,2,1,1,1,3] => 5
100101001 => [1,2,1,1,1,2,1] => 1
100101010 => [1,2,1,1,1,1,1,1] => 1
100101011 => [1,2,1,1,1,1,2] => 1
100101100 => [1,2,1,1,2,2] => 2
100101101 => [1,2,1,1,2,1,1] => 1
100101110 => [1,2,1,1,3,1] => 4
100101111 => [1,2,1,1,4] => 18
100110000 => [1,2,2,4] => 26
100110001 => [1,2,2,3,1] => 5
100110010 => [1,2,2,2,1,1] => 5
100110011 => [1,2,2,2,2] => 14
100110100 => [1,2,2,1,1,2] => 2
100110101 => [1,2,2,1,1,1,1] => 2
100110110 => [1,2,2,1,2,1] => 2
100110111 => [1,2,2,1,3] => 7
100111000 => [1,2,3,3] => 12
100111001 => [1,2,3,2,1] => 7
100111010 => [1,2,3,1,1,1] => 2
100111011 => [1,2,3,1,2] => 2
100111100 => [1,2,4,2] => 28
100111101 => [1,2,4,1,1] => 7
100111110 => [1,2,5,1] => 16
100111111 => [1,2,6] => 30
101000000 => [1,1,1,6] => 35
101000001 => [1,1,1,5,1] => 20
101000010 => [1,1,1,4,1,1] => 10
101000011 => [1,1,1,4,2] => 25
101000100 => [1,1,1,3,1,2] => 4
101000101 => [1,1,1,3,1,1,1] => 4
101000110 => [1,1,1,3,2,1] => 9
101000111 => [1,1,1,3,3] => 32
101001000 => [1,1,1,2,1,3] => 3
101001001 => [1,1,1,2,1,2,1] => 1
101001010 => [1,1,1,2,1,1,1,1] => 1
101001011 => [1,1,1,2,1,1,2] => 1
101001100 => [1,1,1,2,2,2] => 5
101001101 => [1,1,1,2,2,1,1] => 2
101001110 => [1,1,1,2,3,1] => 2
101001111 => [1,1,1,2,4] => 11
101010000 => [1,1,1,1,1,4] => 21
101010001 => [1,1,1,1,1,3,1] => 6
101010010 => [1,1,1,1,1,2,1,1] => 1
101010011 => [1,1,1,1,1,2,2] => 2
101010100 => [1,1,1,1,1,1,1,2] => 1
101010101 => [1,1,1,1,1,1,1,1,1] => 1
101010110 => [1,1,1,1,1,1,2,1] => 1
101010111 => [1,1,1,1,1,1,3] => 7
101011000 => [1,1,1,1,2,3] => 2
101011001 => [1,1,1,1,2,2,1] => 2
101011010 => [1,1,1,1,2,1,1,1] => 1
101011011 => [1,1,1,1,2,1,2] => 1
101011100 => [1,1,1,1,3,2] => 11
101011101 => [1,1,1,1,3,1,1] => 5
101011110 => [1,1,1,1,4,1] => 15
101011111 => [1,1,1,1,5] => 35
101100000 => [1,1,2,5] => 25
101100001 => [1,1,2,4,1] => 9
101100010 => [1,1,2,3,1,1] => 2
101100011 => [1,1,2,3,2] => 7
101100100 => [1,1,2,2,1,2] => 2
101100101 => [1,1,2,2,1,1,1] => 2
101100110 => [1,1,2,2,2,1] => 5
101100111 => [1,1,2,2,3] => 5
101101000 => [1,1,2,1,1,3] => 4
101101001 => [1,1,2,1,1,2,1] => 1
101101010 => [1,1,2,1,1,1,1,1] => 1
101101011 => [1,1,2,1,1,1,2] => 1
101101100 => [1,1,2,1,2,2] => 2
101101101 => [1,1,2,1,2,1,1] => 1
101101110 => [1,1,2,1,3,1] => 3
101101111 => [1,1,2,1,4] => 15
101110000 => [1,1,3,4] => 21
101110001 => [1,1,3,3,1] => 21
101110010 => [1,1,3,2,1,1] => 7
101110011 => [1,1,3,2,2] => 19
101110100 => [1,1,3,1,1,2] => 3
101110101 => [1,1,3,1,1,1,1] => 3
101110110 => [1,1,3,1,2,1] => 3
101110111 => [1,1,3,1,3] => 15
101111000 => [1,1,4,3] => 67
101111001 => [1,1,4,2,1] => 16
101111010 => [1,1,4,1,1,1] => 6
101111011 => [1,1,4,1,2] => 6
101111100 => [1,1,5,2] => 30
101111101 => [1,1,5,1,1] => 10
101111110 => [1,1,6,1] => 15
101111111 => [1,1,7] => 21
110000000 => [2,7] => 20
110000001 => [2,6,1] => 14
110000010 => [2,5,1,1] => 9
110000011 => [2,5,2] => 44
110000100 => [2,4,1,2] => 5
110000101 => [2,4,1,1,1] => 5
110000110 => [2,4,2,1] => 21
110000111 => [2,4,3] => 42
110001000 => [2,3,1,3] => 9
110001001 => [2,3,1,2,1] => 2
110001010 => [2,3,1,1,1,1] => 2
110001011 => [2,3,1,1,2] => 2
110001100 => [2,3,2,2] => 23
110001101 => [2,3,2,1,1] => 7
110001110 => [2,3,3,1] => 12
110001111 => [2,3,4] => 12
110010000 => [2,2,1,4] => 33
110010001 => [2,2,1,3,1] => 7
110010010 => [2,2,1,2,1,1] => 2
110010011 => [2,2,1,2,2] => 4
110010100 => [2,2,1,1,1,2] => 2
110010101 => [2,2,1,1,1,1,1] => 2
110010110 => [2,2,1,1,2,1] => 2
110010111 => [2,2,1,1,3] => 9
110011000 => [2,2,2,3] => 14
110011001 => [2,2,2,2,1] => 14
110011010 => [2,2,2,1,1,1] => 5
110011011 => [2,2,2,1,2] => 5
110011100 => [2,2,3,2] => 19
110011101 => [2,2,3,1,1] => 5
110011110 => [2,2,4,1] => 21
110011111 => [2,2,5] => 56
110100000 => [2,1,1,5] => 34
110100001 => [2,1,1,4,1] => 14
110100010 => [2,1,1,3,1,1] => 4
110100011 => [2,1,1,3,2] => 9
110100100 => [2,1,1,2,1,2] => 1
110100101 => [2,1,1,2,1,1,1] => 1
110100110 => [2,1,1,2,2,1] => 2
110100111 => [2,1,1,2,3] => 2
110101000 => [2,1,1,1,1,3] => 6
110101001 => [2,1,1,1,1,2,1] => 1
110101010 => [2,1,1,1,1,1,1,1] => 1
110101011 => [2,1,1,1,1,1,2] => 1
110101100 => [2,1,1,1,2,2] => 2
110101101 => [2,1,1,1,2,1,1] => 1
110101110 => [2,1,1,1,3,1] => 5
110101111 => [2,1,1,1,4] => 20
110110000 => [2,1,2,4] => 11
110110001 => [2,1,2,3,1] => 2
110110010 => [2,1,2,2,1,1] => 2
110110011 => [2,1,2,2,2] => 5
110110100 => [2,1,2,1,1,2] => 1
110110101 => [2,1,2,1,1,1,1] => 1
110110110 => [2,1,2,1,2,1] => 1
110110111 => [2,1,2,1,3] => 3
110111000 => [2,1,3,3] => 21
110111001 => [2,1,3,2,1] => 7
110111010 => [2,1,3,1,1,1] => 3
110111011 => [2,1,3,1,2] => 3
110111100 => [2,1,4,2] => 23
110111101 => [2,1,4,1,1] => 9
110111110 => [2,1,5,1] => 19
110111111 => [2,1,6] => 34
111000000 => [3,6] => 28
111000001 => [3,5,1] => 14
111000010 => [3,4,1,1] => 5
111000011 => [3,4,2] => 26
111000100 => [3,3,1,2] => 5
111000101 => [3,3,1,1,1] => 5
111000110 => [3,3,2,1] => 21
111000111 => [3,3,3] => 42
111001000 => [3,2,1,3] => 12
111001001 => [3,2,1,2,1] => 3
111001010 => [3,2,1,1,1,1] => 3
111001011 => [3,2,1,1,2] => 3
111001100 => [3,2,2,2] => 28
111001101 => [3,2,2,1,1] => 9
111001110 => [3,2,3,1] => 9
111001111 => [3,2,4] => 30
111010000 => [3,1,1,4] => 14
111010001 => [3,1,1,3,1] => 5
111010010 => [3,1,1,2,1,1] => 1
111010011 => [3,1,1,2,2] => 2
111010100 => [3,1,1,1,1,2] => 1
111010101 => [3,1,1,1,1,1,1] => 1
111010110 => [3,1,1,1,2,1] => 1
111010111 => [3,1,1,1,3] => 6
111011000 => [3,1,2,3] => 2
111011001 => [3,1,2,2,1] => 2
111011010 => [3,1,2,1,1,1] => 1
111011011 => [3,1,2,1,2] => 1
111011100 => [3,1,3,2] => 9
111011101 => [3,1,3,1,1] => 4
111011110 => [3,1,4,1] => 9
111011111 => [3,1,5] => 28
111100000 => [4,5] => 14
111100001 => [4,4,1] => 14
111100010 => [4,3,1,1] => 9
111100011 => [4,3,2] => 44
111100100 => [4,2,1,2] => 4
111100101 => [4,2,1,1,1] => 4
111100110 => [4,2,2,1] => 14
111100111 => [4,2,3] => 14
111101000 => [4,1,1,3] => 6
111101001 => [4,1,1,2,1] => 1
111101010 => [4,1,1,1,1,1] => 1
111101011 => [4,1,1,1,2] => 1
111101100 => [4,1,2,2] => 2
111101101 => [4,1,2,1,1] => 1
111101110 => [4,1,3,1] => 5
111101111 => [4,1,4] => 14
111110000 => [5,4] => 28
111110001 => [5,3,1] => 14
111110010 => [5,2,1,1] => 5
111110011 => [5,2,2] => 20
111110100 => [5,1,1,2] => 1
111110101 => [5,1,1,1,1] => 1
111110110 => [5,1,2,1] => 1
111110111 => [5,1,3] => 6
111111000 => [6,3] => 20
111111001 => [6,2,1] => 6
111111010 => [6,1,1,1] => 1
111111011 => [6,1,2] => 1
111111100 => [7,2] => 7
111111101 => [7,1,1] => 1
111111110 => [8,1] => 1
111111111 => [9] => 1
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Generating function
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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
12,4 18,8,6 28,12,8,6,8,2 42,22,10,10,20,2,6,0,10,4,0,2 62,42,18,12,30,8,12,0,20,4,0,8,0,18,4,4,0,0,2,2,8,0,0,0,0,0,0,0,0,2
$F_{1} = 2\ q$
$F_{2} = 4\ q$
$F_{3} = 8\ q$
$F_{4} = 12\ q + 4\ q^{2}$
$F_{5} = 18\ q + 8\ q^{2} + 6\ q^{3}$
$F_{6} = 28\ q + 12\ q^{2} + 8\ q^{3} + 6\ q^{4} + 8\ q^{5} + 2\ q^{6}$
$F_{7} = 42\ q + 22\ q^{2} + 10\ q^{3} + 10\ q^{4} + 20\ q^{5} + 2\ q^{6} + 6\ q^{7} + 10\ q^{9} + 4\ q^{10} + 2\ q^{12}$
$F_{8} = 62\ q + 42\ q^{2} + 18\ q^{3} + 12\ q^{4} + 30\ q^{5} + 8\ q^{6} + 12\ q^{7} + 20\ q^{9} + 4\ q^{10} + 8\ q^{12} + 18\ q^{14} + 4\ q^{15} + 4\ q^{16} + 2\ q^{19} + 2\ q^{20} + 8\ q^{21} + 2\ q^{30}$
$F_{9} = 92\ q + 74\ q^{2} + 30\ q^{3} + 24\ q^{4} + 46\ q^{5} + 18\ q^{6} + 24\ q^{7} + 28\ q^{9} + 4\ q^{10} + 8\ q^{11} + 14\ q^{12} + 32\ q^{14} + 8\ q^{15} + 4\ q^{16} + 2\ q^{17} + 2\ q^{18} + 6\ q^{19} + 12\ q^{20} + 18\ q^{21} + 4\ q^{23} + 4\ q^{25} + 4\ q^{26} + 12\ q^{28} + 8\ q^{30} + 2\ q^{31} + 2\ q^{32} + 2\ q^{33} + 4\ q^{34} + 4\ q^{35} + 8\ q^{42} + 4\ q^{44} + 4\ q^{56} + 2\ q^{58} + 2\ q^{67}$
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.
See [1, Def. 4.2.6].
Apparently, the total number of tableaux of given size is the number of involutions.
Map
delta morphism
Description
Applies the delta morphism to a binary word.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
The delta morphism of a finite word $w$ is the integer compositions composed of the lengths of consecutive runs of the same letter in $w$.
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