Identifier
-
Mp00234:
Binary words
—valleys-to-peaks⟶
Binary words
St000847: Binary words ⟶ ℤ
Values
0 => 1 => 1
1 => 1 => 1
00 => 01 => 1
01 => 10 => 1
10 => 11 => 1
11 => 11 => 1
000 => 001 => 1
001 => 010 => 2
010 => 101 => 2
011 => 101 => 2
100 => 101 => 2
101 => 110 => 1
110 => 111 => 1
111 => 111 => 1
0000 => 0001 => 1
0001 => 0010 => 2
0010 => 0101 => 3
0011 => 0101 => 3
0100 => 1001 => 2
0101 => 1010 => 3
0110 => 1011 => 2
0111 => 1011 => 2
1000 => 1001 => 2
1001 => 1010 => 3
1010 => 1101 => 2
1011 => 1101 => 2
1100 => 1101 => 2
1101 => 1110 => 1
1110 => 1111 => 1
1111 => 1111 => 1
00000 => 00001 => 1
00001 => 00010 => 2
00010 => 00101 => 3
00011 => 00101 => 3
00100 => 01001 => 4
00101 => 01010 => 6
00110 => 01011 => 3
00111 => 01011 => 3
01000 => 10001 => 2
01001 => 10010 => 4
01010 => 10101 => 6
01011 => 10101 => 6
01100 => 10101 => 6
01101 => 10110 => 4
01110 => 10111 => 2
01111 => 10111 => 2
10000 => 10001 => 2
10001 => 10010 => 4
10010 => 10101 => 6
10011 => 10101 => 6
10100 => 11001 => 2
10101 => 11010 => 3
10110 => 11011 => 3
10111 => 11011 => 3
11000 => 11001 => 2
11001 => 11010 => 3
11010 => 11101 => 2
11011 => 11101 => 2
11100 => 11101 => 2
11101 => 11110 => 1
11110 => 11111 => 1
11111 => 11111 => 1
000000 => 000001 => 1
000001 => 000010 => 2
000010 => 000101 => 3
000011 => 000101 => 3
000100 => 001001 => 5
000101 => 001010 => 7
000110 => 001011 => 3
000111 => 001011 => 3
001000 => 010001 => 4
001001 => 010010 => 8
001010 => 010101 => 11
001011 => 010101 => 11
001100 => 010101 => 11
001101 => 010110 => 7
001110 => 010111 => 3
001111 => 010111 => 3
010000 => 100001 => 2
010001 => 100010 => 4
010010 => 100101 => 7
010011 => 100101 => 7
010100 => 101001 => 7
010101 => 101010 => 11
010110 => 101011 => 7
010111 => 101011 => 7
011000 => 101001 => 7
011001 => 101010 => 11
011010 => 101101 => 8
011011 => 101101 => 8
011100 => 101101 => 8
011101 => 101110 => 4
011110 => 101111 => 2
011111 => 101111 => 2
100000 => 100001 => 2
100001 => 100010 => 4
100010 => 100101 => 7
100011 => 100101 => 7
100100 => 101001 => 7
100101 => 101010 => 11
100110 => 101011 => 7
>>> Load all 1060 entries. <<<100111 => 101011 => 7
101000 => 110001 => 2
101001 => 110010 => 4
101010 => 110101 => 7
101011 => 110101 => 7
101100 => 110101 => 7
101101 => 110110 => 5
101110 => 110111 => 3
101111 => 110111 => 3
110000 => 110001 => 2
110001 => 110010 => 4
110010 => 110101 => 7
110011 => 110101 => 7
110100 => 111001 => 2
110101 => 111010 => 3
110110 => 111011 => 3
110111 => 111011 => 3
111000 => 111001 => 2
111001 => 111010 => 3
111010 => 111101 => 2
111011 => 111101 => 2
111100 => 111101 => 2
111101 => 111110 => 1
111110 => 111111 => 1
111111 => 111111 => 1
0000000 => 0000001 => 1
0000001 => 0000010 => 2
0000010 => 0000101 => 3
0000011 => 0000101 => 3
0000100 => 0001001 => 5
0000101 => 0001010 => 7
0000110 => 0001011 => 3
0000111 => 0001011 => 3
0001000 => 0010001 => 6
0001001 => 0010010 => 11
0001010 => 0010101 => 13
0001011 => 0010101 => 13
0001100 => 0010101 => 13
0001101 => 0010110 => 8
0001110 => 0010111 => 3
0001111 => 0010111 => 3
0010000 => 0100001 => 4
0010001 => 0100010 => 9
0010010 => 0100101 => 15
0010011 => 0100101 => 15
0010100 => 0101001 => 16
0010101 => 0101010 => 24
0010110 => 0101011 => 13
0010111 => 0101011 => 13
0011000 => 0101001 => 16
0011001 => 0101010 => 24
0011010 => 0101101 => 15
0011011 => 0101101 => 15
0011100 => 0101101 => 15
0011101 => 0101110 => 7
0011110 => 0101111 => 3
0011111 => 0101111 => 3
0100000 => 1000001 => 2
0100001 => 1000010 => 4
0100010 => 1000101 => 7
0100011 => 1000101 => 7
0100100 => 1001001 => 11
0100101 => 1001010 => 16
0100110 => 1001011 => 8
0100111 => 1001011 => 8
0101000 => 1010001 => 7
0101001 => 1010010 => 15
0101010 => 1010101 => 24
0101011 => 1010101 => 24
0101100 => 1010101 => 24
0101101 => 1010110 => 16
0101110 => 1010111 => 7
0101111 => 1010111 => 7
0110000 => 1010001 => 7
0110001 => 1010010 => 15
0110010 => 1010101 => 24
0110011 => 1010101 => 24
0110100 => 1011001 => 10
0110101 => 1011010 => 15
0110110 => 1011011 => 11
0110111 => 1011011 => 11
0111000 => 1011001 => 10
0111001 => 1011010 => 15
0111010 => 1011101 => 9
0111011 => 1011101 => 9
0111100 => 1011101 => 9
0111101 => 1011110 => 4
0111110 => 1011111 => 2
0111111 => 1011111 => 2
1000000 => 1000001 => 2
1000001 => 1000010 => 4
1000010 => 1000101 => 7
1000011 => 1000101 => 7
1000100 => 1001001 => 11
1000101 => 1001010 => 16
1000110 => 1001011 => 8
1000111 => 1001011 => 8
1001000 => 1010001 => 7
1001001 => 1010010 => 15
1001010 => 1010101 => 24
1001011 => 1010101 => 24
1001100 => 1010101 => 24
1001101 => 1010110 => 16
1001110 => 1010111 => 7
1001111 => 1010111 => 7
1010000 => 1100001 => 2
1010001 => 1100010 => 4
1010010 => 1100101 => 9
1010011 => 1100101 => 9
1010100 => 1101001 => 8
1010101 => 1101010 => 13
1010110 => 1101011 => 10
1010111 => 1101011 => 10
1011000 => 1101001 => 8
1011001 => 1101010 => 13
1011010 => 1101101 => 11
1011011 => 1101101 => 11
1011100 => 1101101 => 11
1011101 => 1101110 => 6
1011110 => 1101111 => 3
1011111 => 1101111 => 3
1100000 => 1100001 => 2
1100001 => 1100010 => 4
1100010 => 1100101 => 9
1100011 => 1100101 => 9
1100100 => 1101001 => 8
1100101 => 1101010 => 13
1100110 => 1101011 => 10
1100111 => 1101011 => 10
1101000 => 1110001 => 2
1101001 => 1110010 => 4
1101010 => 1110101 => 7
1101011 => 1110101 => 7
1101100 => 1110101 => 7
1101101 => 1110110 => 5
1101110 => 1110111 => 4
1101111 => 1110111 => 4
1110000 => 1110001 => 2
1110001 => 1110010 => 4
1110010 => 1110101 => 7
1110011 => 1110101 => 7
1110100 => 1111001 => 2
1110101 => 1111010 => 3
1110110 => 1111011 => 3
1110111 => 1111011 => 3
1111000 => 1111001 => 2
1111001 => 1111010 => 3
1111010 => 1111101 => 2
1111011 => 1111101 => 2
1111100 => 1111101 => 2
1111101 => 1111110 => 1
1111110 => 1111111 => 1
1111111 => 1111111 => 1
00000000 => 00000001 => 1
00000001 => 00000010 => 2
00000010 => 00000101 => 3
00000011 => 00000101 => 3
00000100 => 00001001 => 5
00000101 => 00001010 => 7
00000110 => 00001011 => 3
00000111 => 00001011 => 3
00001000 => 00010001 => 7
00001001 => 00010010 => 12
00001010 => 00010101 => 13
00001011 => 00010101 => 13
00001100 => 00010101 => 13
00001101 => 00010110 => 8
00001110 => 00010111 => 3
00001111 => 00010111 => 3
00010000 => 00100001 => 6
00010001 => 00100010 => 13
00010010 => 00100101 => 20
00010011 => 00100101 => 20
00010100 => 00101001 => 22
00010101 => 00101010 => 31
00010110 => 00101011 => 15
00010111 => 00101011 => 15
00011000 => 00101001 => 22
00011001 => 00101010 => 31
00011010 => 00101101 => 17
00011011 => 00101101 => 17
00011100 => 00101101 => 17
00011101 => 00101110 => 8
00011110 => 00101111 => 3
00011111 => 00101111 => 3
00100000 => 01000001 => 4
00100001 => 01000010 => 9
00100010 => 01000101 => 16
00100011 => 01000101 => 16
00100100 => 01001001 => 24
00100101 => 01001010 => 35
00100110 => 01001011 => 17
00100111 => 01001011 => 17
00101000 => 01010001 => 17
00101001 => 01010010 => 35
00101010 => 01010101 => 51
00101011 => 01010101 => 51
00101100 => 01010101 => 51
00101101 => 01010110 => 33
00101110 => 01010111 => 13
00101111 => 01010111 => 13
00110000 => 01010001 => 17
00110001 => 01010010 => 35
00110010 => 01010101 => 51
00110011 => 01010101 => 51
00110100 => 01011001 => 23
00110101 => 01011010 => 32
00110110 => 01011011 => 20
00110111 => 01011011 => 20
00111000 => 01011001 => 23
00111001 => 01011010 => 32
00111010 => 01011101 => 16
00111011 => 01011101 => 16
00111100 => 01011101 => 16
00111101 => 01011110 => 7
00111110 => 01011111 => 3
00111111 => 01011111 => 3
01000000 => 10000001 => 2
01000001 => 10000010 => 4
01000010 => 10000101 => 7
01000011 => 10000101 => 7
01000100 => 10001001 => 12
01000101 => 10001010 => 17
01000110 => 10001011 => 8
01000111 => 10001011 => 8
01001000 => 10010001 => 12
01001001 => 10010010 => 24
01001010 => 10010101 => 33
01001011 => 10010101 => 33
01001100 => 10010101 => 33
01001101 => 10010110 => 21
01001110 => 10010111 => 8
01001111 => 10010111 => 8
01010000 => 10100001 => 7
01010001 => 10100010 => 16
01010010 => 10100101 => 32
01010011 => 10100101 => 32
01010100 => 10101001 => 33
01010101 => 10101010 => 51
01010110 => 10101011 => 31
01010111 => 10101011 => 31
01011000 => 10101001 => 33
01011001 => 10101010 => 51
01011010 => 10101101 => 35
01011011 => 10101101 => 35
01011100 => 10101101 => 35
01011101 => 10101110 => 17
01011110 => 10101111 => 7
01011111 => 10101111 => 7
01100000 => 10100001 => 7
01100001 => 10100010 => 16
01100010 => 10100101 => 32
01100011 => 10100101 => 32
01100100 => 10101001 => 33
01100101 => 10101010 => 51
01100110 => 10101011 => 31
01100111 => 10101011 => 31
01101000 => 10110001 => 10
01101001 => 10110010 => 21
01101010 => 10110101 => 35
01101011 => 10110101 => 35
01101100 => 10110101 => 35
01101101 => 10110110 => 24
01101110 => 10110111 => 12
01101111 => 10110111 => 12
01110000 => 10110001 => 10
01110001 => 10110010 => 21
01110010 => 10110101 => 35
01110011 => 10110101 => 35
01110100 => 10111001 => 11
01110101 => 10111010 => 16
01110110 => 10111011 => 13
01110111 => 10111011 => 13
01111000 => 10111001 => 11
01111001 => 10111010 => 16
01111010 => 10111101 => 9
01111011 => 10111101 => 9
01111100 => 10111101 => 9
01111101 => 10111110 => 4
01111110 => 10111111 => 2
01111111 => 10111111 => 2
10000000 => 10000001 => 2
10000001 => 10000010 => 4
10000010 => 10000101 => 7
10000011 => 10000101 => 7
10000100 => 10001001 => 12
10000101 => 10001010 => 17
10000110 => 10001011 => 8
10000111 => 10001011 => 8
10001000 => 10010001 => 12
10001001 => 10010010 => 24
10001010 => 10010101 => 33
10001011 => 10010101 => 33
10001100 => 10010101 => 33
10001101 => 10010110 => 21
10001110 => 10010111 => 8
10001111 => 10010111 => 8
10010000 => 10100001 => 7
10010001 => 10100010 => 16
10010010 => 10100101 => 32
10010011 => 10100101 => 32
10010100 => 10101001 => 33
10010101 => 10101010 => 51
10010110 => 10101011 => 31
10010111 => 10101011 => 31
10011000 => 10101001 => 33
10011001 => 10101010 => 51
10011010 => 10101101 => 35
10011011 => 10101101 => 35
10011100 => 10101101 => 35
10011101 => 10101110 => 17
10011110 => 10101111 => 7
10011111 => 10101111 => 7
10100000 => 11000001 => 2
10100001 => 11000010 => 4
10100010 => 11000101 => 9
10100011 => 11000101 => 9
10100100 => 11001001 => 13
10100101 => 11001010 => 20
10100110 => 11001011 => 12
10100111 => 11001011 => 12
10101000 => 11010001 => 8
10101001 => 11010010 => 17
10101010 => 11010101 => 31
10101011 => 11010101 => 31
10101100 => 11010101 => 31
10101101 => 11010110 => 22
10101110 => 11010111 => 11
10101111 => 11010111 => 11
10110000 => 11010001 => 8
10110001 => 11010010 => 17
10110010 => 11010101 => 31
10110011 => 11010101 => 31
10110100 => 11011001 => 13
10110101 => 11011010 => 20
10110110 => 11011011 => 16
10110111 => 11011011 => 16
10111000 => 11011001 => 13
10111001 => 11011010 => 20
10111010 => 11011101 => 13
10111011 => 11011101 => 13
10111100 => 11011101 => 13
10111101 => 11011110 => 6
10111110 => 11011111 => 3
10111111 => 11011111 => 3
11000000 => 11000001 => 2
11000001 => 11000010 => 4
11000010 => 11000101 => 9
11000011 => 11000101 => 9
11000100 => 11001001 => 13
11000101 => 11001010 => 20
11000110 => 11001011 => 12
11000111 => 11001011 => 12
11001000 => 11010001 => 8
11001001 => 11010010 => 17
11001010 => 11010101 => 31
11001011 => 11010101 => 31
11001100 => 11010101 => 31
11001101 => 11010110 => 22
11001110 => 11010111 => 11
11001111 => 11010111 => 11
11010000 => 11100001 => 2
11010001 => 11100010 => 4
11010010 => 11100101 => 9
11010011 => 11100101 => 9
11010100 => 11101001 => 8
11010101 => 11101010 => 13
11010110 => 11101011 => 11
11010111 => 11101011 => 11
11011000 => 11101001 => 8
11011001 => 11101010 => 13
11011010 => 11101101 => 12
11011011 => 11101101 => 12
11011100 => 11101101 => 12
11011101 => 11101110 => 7
11011110 => 11101111 => 4
11011111 => 11101111 => 4
11100000 => 11100001 => 2
11100001 => 11100010 => 4
11100010 => 11100101 => 9
11100011 => 11100101 => 9
11100100 => 11101001 => 8
11100101 => 11101010 => 13
11100110 => 11101011 => 11
11100111 => 11101011 => 11
11101000 => 11110001 => 2
11101001 => 11110010 => 4
11101010 => 11110101 => 7
11101011 => 11110101 => 7
11101100 => 11110101 => 7
11101101 => 11110110 => 5
11101110 => 11110111 => 4
11101111 => 11110111 => 4
11110000 => 11110001 => 2
11110001 => 11110010 => 4
11110010 => 11110101 => 7
11110011 => 11110101 => 7
11110100 => 11111001 => 2
11110101 => 11111010 => 3
11110110 => 11111011 => 3
11110111 => 11111011 => 3
11111000 => 11111001 => 2
11111001 => 11111010 => 3
11111010 => 11111101 => 2
11111011 => 11111101 => 2
11111100 => 11111101 => 2
11111101 => 11111110 => 1
11111110 => 11111111 => 1
11111111 => 11111111 => 1
000000000 => 000000001 => 1
000000001 => 000000010 => 2
000000010 => 000000101 => 3
000000011 => 000000101 => 3
000000100 => 000001001 => 5
000000101 => 000001010 => 7
000000110 => 000001011 => 3
000000111 => 000001011 => 3
000001000 => 000010001 => 7
000001001 => 000010010 => 12
000001010 => 000010101 => 13
000001011 => 000010101 => 13
000001100 => 000010101 => 13
000001101 => 000010110 => 8
000001110 => 000010111 => 3
000001111 => 000010111 => 3
000010000 => 000100001 => 8
000010001 => 000100010 => 16
000010010 => 000100101 => 22
000010011 => 000100101 => 22
000010100 => 000101001 => 24
000010101 => 000101010 => 33
000010110 => 000101011 => 15
000010111 => 000101011 => 15
000011000 => 000101001 => 24
000011001 => 000101010 => 33
000011010 => 000101101 => 17
000011011 => 000101101 => 17
000011100 => 000101101 => 17
000011101 => 000101110 => 8
000011110 => 000101111 => 3
000011111 => 000101111 => 3
000100000 => 001000001 => 6
000100001 => 001000010 => 14
000100010 => 001000101 => 24
000100011 => 001000101 => 24
000100100 => 001001001 => 35
000100101 => 001001010 => 50
000100110 => 001001011 => 23
000100111 => 001001011 => 23
000101000 => 001010001 => 27
000101001 => 001010010 => 52
000101010 => 001010101 => 67
000101011 => 001010101 => 67
000101100 => 001010101 => 67
000101101 => 001010110 => 42
000101110 => 001010111 => 15
000101111 => 001010111 => 15
000110000 => 001010001 => 27
000110001 => 001010010 => 52
000110010 => 001010101 => 67
000110011 => 001010101 => 67
000110100 => 001011001 => 31
000110101 => 001011010 => 41
000110110 => 001011011 => 23
000110111 => 001011011 => 23
000111000 => 001011001 => 31
000111001 => 001011010 => 41
000111010 => 001011101 => 18
000111011 => 001011101 => 18
000111100 => 001011101 => 18
000111101 => 001011110 => 8
000111110 => 001011111 => 3
000111111 => 001011111 => 3
001000000 => 010000001 => 4
001000001 => 010000010 => 9
001000010 => 010000101 => 16
001000011 => 010000101 => 16
001000100 => 010001001 => 28
001000101 => 010001010 => 40
001000110 => 010001011 => 18
001000111 => 010001011 => 18
001001000 => 010010001 => 29
001001001 => 010010010 => 57
001001010 => 010010101 => 75
001001011 => 010010101 => 75
001001100 => 010010101 => 75
001001101 => 010010110 => 47
001001110 => 010010111 => 17
001001111 => 010010111 => 17
001010000 => 010100001 => 17
001010001 => 010100010 => 40
001010010 => 010100101 => 77
001010011 => 010100101 => 77
001010100 => 010101001 => 80
001010101 => 010101010 => 120
001010110 => 010101011 => 67
001010111 => 010101011 => 67
001011000 => 010101001 => 80
001011001 => 010101010 => 120
001011010 => 010101101 => 75
001011011 => 010101101 => 75
001011100 => 010101101 => 75
001011101 => 010101110 => 35
001011110 => 010101111 => 13
001011111 => 010101111 => 13
001100000 => 010100001 => 17
001100001 => 010100010 => 40
001100010 => 010100101 => 77
001100011 => 010100101 => 77
001100100 => 010101001 => 80
001100101 => 010101010 => 120
001100110 => 010101011 => 67
001100111 => 010101011 => 67
001101000 => 010110001 => 25
001101001 => 010110010 => 51
001101010 => 010110101 => 77
001101011 => 010110101 => 77
001101100 => 010110101 => 77
001101101 => 010110110 => 51
001101110 => 010110111 => 22
001101111 => 010110111 => 22
001110000 => 010110001 => 25
001110001 => 010110010 => 51
001110010 => 010110101 => 77
001110011 => 010110101 => 77
001110100 => 010111001 => 25
001110101 => 010111010 => 34
001110110 => 010111011 => 24
001110111 => 010111011 => 24
001111000 => 010111001 => 25
001111001 => 010111010 => 34
001111010 => 010111101 => 16
001111011 => 010111101 => 16
001111100 => 010111101 => 16
001111101 => 010111110 => 7
001111110 => 010111111 => 3
001111111 => 010111111 => 3
010000000 => 100000001 => 2
010000001 => 100000010 => 4
010000010 => 100000101 => 7
010000011 => 100000101 => 7
010000100 => 100001001 => 12
010000101 => 100001010 => 17
010000110 => 100001011 => 8
010000111 => 100001011 => 8
010001000 => 100010001 => 16
010001001 => 100010010 => 29
010001010 => 100010101 => 35
010001011 => 100010101 => 35
010001100 => 100010101 => 35
010001101 => 100010110 => 22
010001110 => 100010111 => 8
010001111 => 100010111 => 8
010010000 => 100100001 => 12
010010001 => 100100010 => 28
010010010 => 100100101 => 51
010010011 => 100100101 => 51
010010100 => 100101001 => 55
010010101 => 100101010 => 80
010010110 => 100101011 => 42
010010111 => 100101011 => 42
010011000 => 100101001 => 55
010011001 => 100101010 => 80
010011010 => 100101101 => 47
010011011 => 100101101 => 47
010011100 => 100101101 => 47
010011101 => 100101110 => 22
010011110 => 100101111 => 8
010011111 => 100101111 => 8
010100000 => 101000001 => 7
010100001 => 101000010 => 16
010100010 => 101000101 => 34
010100011 => 101000101 => 34
010100100 => 101001001 => 51
010100101 => 101001010 => 77
010100110 => 101001011 => 41
010100111 => 101001011 => 41
010101000 => 101010001 => 35
010101001 => 101010010 => 75
010101010 => 101010101 => 120
010101011 => 101010101 => 120
010101100 => 101010101 => 120
010101101 => 101010110 => 80
010101110 => 101010111 => 33
010101111 => 101010111 => 33
010110000 => 101010001 => 35
010110001 => 101010010 => 75
010110010 => 101010101 => 120
010110011 => 101010101 => 120
010110100 => 101011001 => 54
010110101 => 101011010 => 77
010110110 => 101011011 => 50
010110111 => 101011011 => 50
010111000 => 101011001 => 54
010111001 => 101011010 => 77
010111010 => 101011101 => 40
010111011 => 101011101 => 40
010111100 => 101011101 => 40
010111101 => 101011110 => 17
010111110 => 101011111 => 7
010111111 => 101011111 => 7
011000000 => 101000001 => 7
011000001 => 101000010 => 16
011000010 => 101000101 => 34
011000011 => 101000101 => 34
011000100 => 101001001 => 51
011000101 => 101001010 => 77
011000110 => 101001011 => 41
011000111 => 101001011 => 41
011001000 => 101010001 => 35
011001001 => 101010010 => 75
011001010 => 101010101 => 120
011001011 => 101010101 => 120
011001100 => 101010101 => 120
011001101 => 101010110 => 80
011001110 => 101010111 => 33
011001111 => 101010111 => 33
011010000 => 101100001 => 10
011010001 => 101100010 => 23
011010010 => 101100101 => 51
011010011 => 101100101 => 51
011010100 => 101101001 => 47
011010101 => 101101010 => 75
011010110 => 101101011 => 52
011010111 => 101101011 => 52
011011000 => 101101001 => 47
011011001 => 101101010 => 75
011011010 => 101101101 => 57
011011011 => 101101101 => 57
011011100 => 101101101 => 57
011011101 => 101101110 => 29
011011110 => 101101111 => 12
011011111 => 101101111 => 12
011100000 => 101100001 => 10
011100001 => 101100010 => 23
011100010 => 101100101 => 51
011100011 => 101100101 => 51
011100100 => 101101001 => 47
011100101 => 101101010 => 75
011100110 => 101101011 => 52
011100111 => 101101011 => 52
011101000 => 101110001 => 11
011101001 => 101110010 => 23
011101010 => 101110101 => 40
011101011 => 101110101 => 40
011101100 => 101110101 => 40
011101101 => 101110110 => 28
011101110 => 101110111 => 16
011101111 => 101110111 => 16
011110000 => 101110001 => 11
011110001 => 101110010 => 23
011110010 => 101110101 => 40
011110011 => 101110101 => 40
011110100 => 101111001 => 11
011110101 => 101111010 => 16
011110110 => 101111011 => 14
011110111 => 101111011 => 14
011111000 => 101111001 => 11
011111001 => 101111010 => 16
011111010 => 101111101 => 9
011111011 => 101111101 => 9
011111100 => 101111101 => 9
011111101 => 101111110 => 4
011111110 => 101111111 => 2
011111111 => 101111111 => 2
100000000 => 100000001 => 2
100000001 => 100000010 => 4
100000010 => 100000101 => 7
100000011 => 100000101 => 7
100000100 => 100001001 => 12
100000101 => 100001010 => 17
100000110 => 100001011 => 8
100000111 => 100001011 => 8
100001000 => 100010001 => 16
100001001 => 100010010 => 29
100001010 => 100010101 => 35
100001011 => 100010101 => 35
100001100 => 100010101 => 35
100001101 => 100010110 => 22
100001110 => 100010111 => 8
100001111 => 100010111 => 8
100010000 => 100100001 => 12
100010001 => 100100010 => 28
100010010 => 100100101 => 51
100010011 => 100100101 => 51
100010100 => 100101001 => 55
100010101 => 100101010 => 80
100010110 => 100101011 => 42
100010111 => 100101011 => 42
100011000 => 100101001 => 55
100011001 => 100101010 => 80
100011010 => 100101101 => 47
100011011 => 100101101 => 47
100011100 => 100101101 => 47
100011101 => 100101110 => 22
100011110 => 100101111 => 8
100011111 => 100101111 => 8
100100000 => 101000001 => 7
100100001 => 101000010 => 16
100100010 => 101000101 => 34
100100011 => 101000101 => 34
100100100 => 101001001 => 51
100100101 => 101001010 => 77
100100110 => 101001011 => 41
100100111 => 101001011 => 41
100101000 => 101010001 => 35
100101001 => 101010010 => 75
100101010 => 101010101 => 120
100101011 => 101010101 => 120
100101100 => 101010101 => 120
100101101 => 101010110 => 80
100101110 => 101010111 => 33
100101111 => 101010111 => 33
100110000 => 101010001 => 35
100110001 => 101010010 => 75
100110010 => 101010101 => 120
100110011 => 101010101 => 120
100110100 => 101011001 => 54
100110101 => 101011010 => 77
100110110 => 101011011 => 50
100110111 => 101011011 => 50
100111000 => 101011001 => 54
100111001 => 101011010 => 77
100111010 => 101011101 => 40
100111011 => 101011101 => 40
100111100 => 101011101 => 40
100111101 => 101011110 => 17
100111110 => 101011111 => 7
100111111 => 101011111 => 7
101000000 => 110000001 => 2
101000001 => 110000010 => 4
101000010 => 110000101 => 9
101000011 => 110000101 => 9
101000100 => 110001001 => 15
101000101 => 110001010 => 22
101000110 => 110001011 => 12
101000111 => 110001011 => 12
101001000 => 110010001 => 14
101001001 => 110010010 => 29
101001010 => 110010101 => 47
101001011 => 110010101 => 47
101001100 => 110010101 => 47
101001101 => 110010110 => 32
101001110 => 110010111 => 13
101001111 => 110010111 => 13
101010000 => 110100001 => 8
101010001 => 110100010 => 18
101010010 => 110100101 => 41
101010011 => 110100101 => 41
101010100 => 110101001 => 42
101010101 => 110101010 => 67
101010110 => 110101011 => 46
101010111 => 110101011 => 46
101011000 => 110101001 => 42
101011001 => 110101010 => 67
101011010 => 110101101 => 52
101011011 => 110101101 => 52
101011100 => 110101101 => 52
101011101 => 110101110 => 27
101011110 => 110101111 => 11
101011111 => 110101111 => 11
101100000 => 110100001 => 8
101100001 => 110100010 => 18
101100010 => 110100101 => 41
101100011 => 110100101 => 41
101100100 => 110101001 => 42
101100101 => 110101010 => 67
101100110 => 110101011 => 46
101100111 => 110101011 => 46
101101000 => 110110001 => 13
101101001 => 110110010 => 28
101101010 => 110110101 => 50
101101011 => 110110101 => 50
101101100 => 110110101 => 50
101101101 => 110110110 => 35
101101110 => 110110111 => 19
101101111 => 110110111 => 19
101110000 => 110110001 => 13
101110001 => 110110010 => 28
101110010 => 110110101 => 50
101110011 => 110110101 => 50
101110100 => 110111001 => 16
101110101 => 110111010 => 24
101110110 => 110111011 => 20
101110111 => 110111011 => 20
101111000 => 110111001 => 16
101111001 => 110111010 => 24
101111010 => 110111101 => 14
101111011 => 110111101 => 14
101111100 => 110111101 => 14
101111101 => 110111110 => 6
101111110 => 110111111 => 3
101111111 => 110111111 => 3
110000000 => 110000001 => 2
110000001 => 110000010 => 4
110000010 => 110000101 => 9
110000011 => 110000101 => 9
110000100 => 110001001 => 15
110000101 => 110001010 => 22
110000110 => 110001011 => 12
110000111 => 110001011 => 12
110001000 => 110010001 => 14
110001001 => 110010010 => 29
110001010 => 110010101 => 47
110001011 => 110010101 => 47
110001100 => 110010101 => 47
110001101 => 110010110 => 32
110001110 => 110010111 => 13
110001111 => 110010111 => 13
110010000 => 110100001 => 8
110010001 => 110100010 => 18
110010010 => 110100101 => 41
110010011 => 110100101 => 41
110010100 => 110101001 => 42
110010101 => 110101010 => 67
110010110 => 110101011 => 46
110010111 => 110101011 => 46
110011000 => 110101001 => 42
110011001 => 110101010 => 67
110011010 => 110101101 => 52
110011011 => 110101101 => 52
110011100 => 110101101 => 52
110011101 => 110101110 => 27
110011110 => 110101111 => 11
110011111 => 110101111 => 11
110100000 => 111000001 => 2
110100001 => 111000010 => 4
110100010 => 111000101 => 9
110100011 => 111000101 => 9
110100100 => 111001001 => 13
110100101 => 111001010 => 20
110100110 => 111001011 => 14
110100111 => 111001011 => 14
110101000 => 111010001 => 8
110101001 => 111010010 => 17
110101010 => 111010101 => 33
110101011 => 111010101 => 33
110101100 => 111010101 => 33
110101101 => 111010110 => 24
110101110 => 111010111 => 14
110101111 => 111010111 => 14
110110000 => 111010001 => 8
110110001 => 111010010 => 17
110110010 => 111010101 => 33
110110011 => 111010101 => 33
110110100 => 111011001 => 14
110110101 => 111011010 => 22
110110110 => 111011011 => 19
110110111 => 111011011 => 19
110111000 => 111011001 => 14
110111001 => 111011010 => 22
110111010 => 111011101 => 16
110111011 => 111011101 => 16
110111100 => 111011101 => 16
110111101 => 111011110 => 8
110111110 => 111011111 => 4
110111111 => 111011111 => 4
111000000 => 111000001 => 2
111000001 => 111000010 => 4
111000010 => 111000101 => 9
111000011 => 111000101 => 9
111000100 => 111001001 => 13
111000101 => 111001010 => 20
111000110 => 111001011 => 14
111000111 => 111001011 => 14
111001000 => 111010001 => 8
111001001 => 111010010 => 17
111001010 => 111010101 => 33
111001011 => 111010101 => 33
111001100 => 111010101 => 33
111001101 => 111010110 => 24
111001110 => 111010111 => 14
111001111 => 111010111 => 14
111010000 => 111100001 => 2
111010001 => 111100010 => 4
111010010 => 111100101 => 9
111010011 => 111100101 => 9
111010100 => 111101001 => 8
111010101 => 111101010 => 13
111010110 => 111101011 => 11
111010111 => 111101011 => 11
111011000 => 111101001 => 8
111011001 => 111101010 => 13
111011010 => 111101101 => 12
111011011 => 111101101 => 12
111011100 => 111101101 => 12
111011101 => 111101110 => 7
111011110 => 111101111 => 5
111011111 => 111101111 => 5
111100000 => 111100001 => 2
111100001 => 111100010 => 4
111100010 => 111100101 => 9
111100011 => 111100101 => 9
111100100 => 111101001 => 8
111100101 => 111101010 => 13
111100110 => 111101011 => 11
111100111 => 111101011 => 11
111101000 => 111110001 => 2
111101001 => 111110010 => 4
111101010 => 111110101 => 7
111101011 => 111110101 => 7
111101100 => 111110101 => 7
111101101 => 111110110 => 5
111101110 => 111110111 => 4
111101111 => 111110111 => 4
111110000 => 111110001 => 2
111110001 => 111110010 => 4
111110010 => 111110101 => 7
111110011 => 111110101 => 7
111110100 => 111111001 => 2
111110101 => 111111010 => 3
111110110 => 111111011 => 3
111110111 => 111111011 => 3
111111000 => 111111001 => 2
111111001 => 111111010 => 3
111111010 => 111111101 => 2
111111011 => 111111101 => 2
111111100 => 111111101 => 2
111111101 => 111111110 => 1
111111110 => 111111111 => 1
111111111 => 111111111 => 1
0000000001 => 0000000010 => 2
0001100111 => 0010101011 => 87
0001011101 => 0010101110 => 44
0001011011 => 0010101101 => 96
0001010111 => 0010101011 => 87
0001001111 => 0010010111 => 23
0000111101 => 0001011110 => 8
0000111011 => 0001011101 => 18
0000110111 => 0001011011 => 23
0000101111 => 0001010111 => 15
0000011111 => 0000101111 => 3
0000000000 => 0000000001 => 1
0000000110 => 0000001011 => 3
0000011000 => 0000101001 => 24
0000011110 => 0000101111 => 3
0000010010 => 0000100101 => 22
0001100000 => 0010100001 => 28
0001100110 => 0010101011 => 87
0001001000 => 0010010001 => 45
0001001110 => 0010010111 => 23
0001000010 => 0010000101 => 25
0001011010 => 0010101101 => 96
0001010100 => 0010101001 => 117
0000010100 => 0000101001 => 24
0000000010 => 0000000101 => 3
0000001110 => 0000010111 => 3
0000111110 => 0001011111 => 3
0000000100 => 0000001001 => 5
0000001010 => 0000010101 => 13
0000010110 => 0000101011 => 15
0000101110 => 0001010111 => 15
0001011110 => 0010101111 => 15
0000000101 => 0000001010 => 7
0000001101 => 0000010110 => 8
0000011101 => 0000101110 => 8
0000001001 => 0000010010 => 12
0000010101 => 0000101010 => 33
0000101101 => 0001010110 => 44
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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:
4,4 4,8,4 4,10,8,4,0,6 4,12,12,6,2,0,18,4,0,0,6 4,14,14,10,2,2,18,8,8,6,8,0,8,0,10,6,0,0,0,0,0,0,0,10 4,16,16,14,2,2,20,16,12,2,10,14,18,0,2,12,14,0,0,8,4,4,2,4,0,0,0,0,0,0,16,8,12,0,14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10
$F_{1} = 2\ q$
$F_{2} = 4\ q$
$F_{3} = 4\ q + 4\ q^{2}$
$F_{4} = 4\ q + 8\ q^{2} + 4\ q^{3}$
$F_{5} = 4\ q + 10\ q^{2} + 8\ q^{3} + 4\ q^{4} + 6\ q^{6}$
$F_{6} = 4\ q + 12\ q^{2} + 12\ q^{3} + 6\ q^{4} + 2\ q^{5} + 18\ q^{7} + 4\ q^{8} + 6\ q^{11}$
$F_{7} = 4\ q + 14\ q^{2} + 14\ q^{3} + 10\ q^{4} + 2\ q^{5} + 2\ q^{6} + 18\ q^{7} + 8\ q^{8} + 8\ q^{9} + 6\ q^{10} + 8\ q^{11} + 8\ q^{13} + 10\ q^{15} + 6\ q^{16} + 10\ q^{24}$
$F_{8} = 4\ q + 16\ q^{2} + 16\ q^{3} + 14\ q^{4} + 2\ q^{5} + 2\ q^{6} + 20\ q^{7} + 16\ q^{8} + 12\ q^{9} + 2\ q^{10} + 10\ q^{11} + 14\ q^{12} + 18\ q^{13} + 2\ q^{15} + 12\ q^{16} + 14\ q^{17} + 8\ q^{20} + 4\ q^{21} + 4\ q^{22} + 2\ q^{23} + 4\ q^{24} + 16\ q^{31} + 8\ q^{32} + 12\ q^{33} + 14\ q^{35} + 10\ q^{51}$
$F_{9} = 4\ q + 18\ q^{2} + 18\ q^{3} + 16\ q^{4} + 4\ q^{5} + 2\ q^{6} + 20\ q^{7} + 26\ q^{8} + 16\ q^{9} + 2\ q^{10} + 12\ q^{11} + 14\ q^{12} + 16\ q^{13} + 18\ q^{14} + 6\ q^{15} + 20\ q^{16} + 14\ q^{17} + 8\ q^{18} + 4\ q^{19} + 4\ q^{20} + 12\ q^{22} + 8\ q^{23} + 10\ q^{24} + 4\ q^{25} + 4\ q^{27} + 6\ q^{28} + 6\ q^{29} + 2\ q^{31} + 2\ q^{32} + 16\ q^{33} + 8\ q^{34} + 14\ q^{35} + 14\ q^{40} + 14\ q^{41} + 10\ q^{42} + 6\ q^{46} + 16\ q^{47} + 10\ q^{50} + 14\ q^{51} + 12\ q^{52} + 4\ q^{54} + 4\ q^{55} + 4\ q^{57} + 14\ q^{67} + 14\ q^{75} + 16\ q^{77} + 10\ q^{80} + 16\ q^{120}$
Description
The number of standard Young tableaux whose descent set is the binary word.
A descent in a standard Young tableau is an entry $i$ such that $i+1$ appears in a lower row in English notation.
For example, the tableaux $[[1,2,4],[3]]$ and $[[1,2],[3,4]]$ are those with descent set $\{2\}$, corresponding to the binary word $010$.
A descent in a standard Young tableau is an entry $i$ such that $i+1$ appears in a lower row in English notation.
For example, the tableaux $[[1,2,4],[3]]$ and $[[1,2],[3,4]]$ are those with descent set $\{2\}$, corresponding to the binary word $010$.
Map
valleys-to-peaks
Description
Return the binary word with every valley replaced by a peak.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. This map replaces every valley with a peak.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. This map replaces every valley with a peak.
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