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-----------------------------------------------------------------------------
Statistic identifier: St001313
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
Description: The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$.
See [[St001312]] for this statistic on compositions treated as bounce paths.
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References:
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Code:
def lattice_paths_above_boundary(B):
B = list(B)
if sum(B) == 0:
return [B]
else:
paths = [ B[:1] + p for p in lattice_paths_above_boundary(B[1:]) ]
if B[0] == 0:
i = B.index(1)
B = B[:i] + B[i+1:]
paths.extend( [ [1] + p for p in lattice_paths_above_boundary(B) ] )
return paths
def statistic(w):
return len(lattice_paths_above_boundary(w))
-----------------------------------------------------------------------------
Statistic values:
0 => 1
1 => 1
00 => 1
01 => 2
10 => 1
11 => 1
000 => 1
001 => 3
010 => 2
011 => 3
100 => 1
101 => 2
110 => 1
111 => 1
0000 => 1
0001 => 4
0010 => 3
0011 => 6
0100 => 2
0101 => 5
0110 => 3
0111 => 4
1000 => 1
1001 => 3
1010 => 2
1011 => 3
1100 => 1
1101 => 2
1110 => 1
1111 => 1
00000 => 1
00001 => 5
00010 => 4
00011 => 10
00100 => 3
00101 => 9
00110 => 6
00111 => 10
01000 => 2
01001 => 7
01010 => 5
01011 => 9
01100 => 3
01101 => 7
01110 => 4
01111 => 5
10000 => 1
10001 => 4
10010 => 3
10011 => 6
10100 => 2
10101 => 5
10110 => 3
10111 => 4
11000 => 1
11001 => 3
11010 => 2
11011 => 3
11100 => 1
11101 => 2
11110 => 1
11111 => 1
000000 => 1
000001 => 6
000010 => 5
000011 => 15
000100 => 4
000101 => 14
000110 => 10
000111 => 20
001000 => 3
001001 => 12
001010 => 9
001011 => 19
001100 => 6
001101 => 16
001110 => 10
001111 => 15
010000 => 2
010001 => 9
010010 => 7
010011 => 16
010100 => 5
010101 => 14
010110 => 9
010111 => 14
011000 => 3
011001 => 10
011010 => 7
011011 => 12
011100 => 4
011101 => 9
011110 => 5
011111 => 6
100000 => 1
100001 => 5
100010 => 4
100011 => 10
100100 => 3
100101 => 9
100110 => 6
100111 => 10
101000 => 2
101001 => 7
101010 => 5
101011 => 9
101100 => 3
101101 => 7
101110 => 4
101111 => 5
110000 => 1
110001 => 4
110010 => 3
110011 => 6
110100 => 2
110101 => 5
110110 => 3
110111 => 4
111000 => 1
111001 => 3
111010 => 2
111011 => 3
111100 => 1
111101 => 2
111110 => 1
111111 => 1
0000000 => 1
0000001 => 7
0000010 => 6
0000011 => 21
0000100 => 5
0000101 => 20
0000110 => 15
0000111 => 35
0001000 => 4
0001001 => 18
0001010 => 14
0001011 => 34
0001100 => 10
0001101 => 30
0001110 => 20
0001111 => 35
0010000 => 3
0010001 => 15
0010010 => 12
0010011 => 31
0010100 => 9
0010101 => 28
0010110 => 19
0010111 => 34
0011000 => 6
0011001 => 22
0011010 => 16
0011011 => 31
0011100 => 10
0011101 => 25
0011110 => 15
0011111 => 21
0100000 => 2
0100001 => 11
0100010 => 9
0100011 => 25
0100100 => 7
0100101 => 23
0100110 => 16
0100111 => 30
0101000 => 5
0101001 => 19
0101010 => 14
0101011 => 28
0101100 => 9
0101101 => 23
0101110 => 14
0101111 => 20
0110000 => 3
0110001 => 13
0110010 => 10
0110011 => 22
0110100 => 7
0110101 => 19
0110110 => 12
0110111 => 18
0111000 => 4
0111001 => 13
0111010 => 9
0111011 => 15
0111100 => 5
0111101 => 11
0111110 => 6
0111111 => 7
1000000 => 1
1000001 => 6
1000010 => 5
1000011 => 15
1000100 => 4
1000101 => 14
1000110 => 10
1000111 => 20
1001000 => 3
1001001 => 12
1001010 => 9
1001011 => 19
1001100 => 6
1001101 => 16
1001110 => 10
1001111 => 15
1010000 => 2
1010001 => 9
1010010 => 7
1010011 => 16
1010100 => 5
1010101 => 14
1010110 => 9
1010111 => 14
1011000 => 3
1011001 => 10
1011010 => 7
1011011 => 12
1011100 => 4
1011101 => 9
1011110 => 5
1011111 => 6
1100000 => 1
1100001 => 5
1100010 => 4
1100011 => 10
1100100 => 3
1100101 => 9
1100110 => 6
1100111 => 10
1101000 => 2
1101001 => 7
1101010 => 5
1101011 => 9
1101100 => 3
1101101 => 7
1101110 => 4
1101111 => 5
1110000 => 1
1110001 => 4
1110010 => 3
1110011 => 6
1110100 => 2
1110101 => 5
1110110 => 3
1110111 => 4
1111000 => 1
1111001 => 3
1111010 => 2
1111011 => 3
1111100 => 1
1111101 => 2
1111110 => 1
1111111 => 1
00000000 => 1
00000001 => 8
00000010 => 7
00000011 => 28
00000100 => 6
00000101 => 27
00000110 => 21
00000111 => 56
00001000 => 5
00001001 => 25
00001010 => 20
00001011 => 55
00001100 => 15
00001101 => 50
00001110 => 35
00001111 => 70
00010000 => 4
00010001 => 22
00010010 => 18
00010011 => 52
00010100 => 14
00010101 => 48
00010110 => 34
00010111 => 69
00011000 => 10
00011001 => 40
00011010 => 30
00011011 => 65
00011100 => 20
00011101 => 55
00011110 => 35
00011111 => 56
00100000 => 3
00100001 => 18
00100010 => 15
00100011 => 46
00100100 => 12
00100101 => 43
00100110 => 31
00100111 => 65
00101000 => 9
00101001 => 37
00101010 => 28
00101011 => 62
00101100 => 19
00101101 => 53
00101110 => 34
00101111 => 55
00110000 => 6
00110001 => 28
00110010 => 22
00110011 => 53
00110100 => 16
00110101 => 47
00110110 => 31
00110111 => 52
00111000 => 10
00111001 => 35
00111010 => 25
00111011 => 46
00111100 => 15
00111101 => 36
00111110 => 21
00111111 => 28
01000000 => 2
01000001 => 13
01000010 => 11
01000011 => 36
01000100 => 9
01000101 => 34
01000110 => 25
01000111 => 55
01001000 => 7
01001001 => 30
01001010 => 23
01001011 => 53
01001100 => 16
01001101 => 46
01001110 => 30
01001111 => 50
01010000 => 5
01010001 => 24
01010010 => 19
01010011 => 47
01010100 => 14
01010101 => 42
01010110 => 28
01010111 => 48
01011000 => 9
01011001 => 32
01011010 => 23
01011011 => 43
01011100 => 14
01011101 => 34
01011110 => 20
01011111 => 27
01100000 => 3
01100001 => 16
01100010 => 13
01100011 => 35
01100100 => 10
01100101 => 32
01100110 => 22
01100111 => 40
01101000 => 7
01101001 => 26
01101010 => 19
01101011 => 37
01101100 => 12
01101101 => 30
01101110 => 18
01101111 => 25
01110000 => 4
01110001 => 17
01110010 => 13
01110011 => 28
01110100 => 9
01110101 => 24
01110110 => 15
01110111 => 22
01111000 => 5
01111001 => 16
01111010 => 11
01111011 => 18
01111100 => 6
01111101 => 13
01111110 => 7
01111111 => 8
10000000 => 1
10000001 => 7
10000010 => 6
10000011 => 21
10000100 => 5
10000101 => 20
10000110 => 15
10000111 => 35
10001000 => 4
10001001 => 18
10001010 => 14
10001011 => 34
10001100 => 10
10001101 => 30
10001110 => 20
10001111 => 35
10010000 => 3
10010001 => 15
10010010 => 12
10010011 => 31
10010100 => 9
10010101 => 28
10010110 => 19
10010111 => 34
10011000 => 6
10011001 => 22
10011010 => 16
10011011 => 31
10011100 => 10
10011101 => 25
10011110 => 15
10011111 => 21
10100000 => 2
10100001 => 11
10100010 => 9
10100011 => 25
10100100 => 7
10100101 => 23
10100110 => 16
10100111 => 30
10101000 => 5
10101001 => 19
10101010 => 14
10101011 => 28
10101100 => 9
10101101 => 23
10101110 => 14
10101111 => 20
10110000 => 3
10110001 => 13
10110010 => 10
10110011 => 22
10110100 => 7
10110101 => 19
10110110 => 12
10110111 => 18
10111000 => 4
10111001 => 13
10111010 => 9
10111011 => 15
10111100 => 5
10111101 => 11
10111110 => 6
10111111 => 7
11000000 => 1
11000001 => 6
11000010 => 5
11000011 => 15
11000100 => 4
11000101 => 14
11000110 => 10
11000111 => 20
11001000 => 3
11001001 => 12
11001010 => 9
11001011 => 19
11001100 => 6
11001101 => 16
11001110 => 10
11001111 => 15
11010000 => 2
11010001 => 9
11010010 => 7
11010011 => 16
11010100 => 5
11010101 => 14
11010110 => 9
11010111 => 14
11011000 => 3
11011001 => 10
11011010 => 7
11011011 => 12
11011100 => 4
11011101 => 9
11011110 => 5
11011111 => 6
11100000 => 1
11100001 => 5
11100010 => 4
11100011 => 10
11100100 => 3
11100101 => 9
11100110 => 6
11100111 => 10
11101000 => 2
11101001 => 7
11101010 => 5
11101011 => 9
11101100 => 3
11101101 => 7
11101110 => 4
11101111 => 5
11110000 => 1
11110001 => 4
11110010 => 3
11110011 => 6
11110100 => 2
11110101 => 5
11110110 => 3
11110111 => 4
11111000 => 1
11111001 => 3
11111010 => 2
11111011 => 3
11111100 => 1
11111101 => 2
11111110 => 1
11111111 => 1
000000000 => 1
000000001 => 9
000000010 => 8
000000011 => 36
000000100 => 7
000000101 => 35
000000110 => 28
000000111 => 84
000001000 => 6
000001001 => 33
000001010 => 27
000001011 => 83
000001100 => 21
000001101 => 77
000001110 => 56
000001111 => 126
000010000 => 5
000010001 => 30
000010010 => 25
000010011 => 80
000010100 => 20
000010101 => 75
000010110 => 55
000010111 => 125
000011000 => 15
000011001 => 65
000011010 => 50
000011011 => 120
000011100 => 35
000011101 => 105
000011110 => 70
000011111 => 126
000100000 => 4
000100001 => 26
000100010 => 22
000100011 => 74
000100100 => 18
000100101 => 70
000100110 => 52
000100111 => 121
000101000 => 14
000101001 => 62
000101010 => 48
000101011 => 117
000101100 => 34
000101101 => 103
000101110 => 69
000101111 => 125
000110000 => 10
000110001 => 50
000110010 => 40
000110011 => 105
000110100 => 30
000110101 => 95
000110110 => 65
000110111 => 121
000111000 => 20
000111001 => 75
000111010 => 55
000111011 => 111
000111100 => 35
000111101 => 91
000111110 => 56
000111111 => 84
001000000 => 3
001000001 => 21
001000010 => 18
001000011 => 64
001000100 => 15
001000101 => 61
001000110 => 46
001000111 => 111
001001000 => 12
001001001 => 55
001001010 => 43
001001011 => 108
001001100 => 31
001001101 => 96
001001110 => 65
001001111 => 120
001010000 => 9
001010001 => 46
001010010 => 37
001010011 => 99
001010100 => 28
001010101 => 90
001010110 => 62
001010111 => 117
001011000 => 19
001011001 => 72
001011010 => 53
001011011 => 108
001011100 => 34
001011101 => 89
001011110 => 55
001011111 => 83
001100000 => 6
001100001 => 34
001100010 => 28
001100011 => 81
001100100 => 22
001100101 => 75
001100110 => 53
001100111 => 105
001101000 => 16
001101001 => 63
001101010 => 47
001101011 => 99
001101100 => 31
001101101 => 83
001101110 => 52
001101111 => 80
001110000 => 10
001110001 => 45
001110010 => 35
001110011 => 81
001110100 => 25
001110101 => 71
001110110 => 46
001110111 => 74
001111000 => 15
001111001 => 51
001111010 => 36
001111011 => 64
001111100 => 21
001111101 => 49
001111110 => 28
001111111 => 36
010000000 => 2
010000001 => 15
010000010 => 13
010000011 => 49
010000100 => 11
010000101 => 47
010000110 => 36
010000111 => 91
010001000 => 9
010001001 => 43
010001010 => 34
010001011 => 89
010001100 => 25
010001101 => 80
010001110 => 55
010001111 => 105
010010000 => 7
010010001 => 37
010010010 => 30
010010011 => 83
010010100 => 23
010010101 => 76
010010110 => 53
010010111 => 103
010011000 => 16
010011001 => 62
010011010 => 46
010011011 => 96
010011100 => 30
010011101 => 80
010011110 => 50
010011111 => 77
010100000 => 5
010100001 => 29
010100010 => 24
010100011 => 71
010100100 => 19
010100101 => 66
010100110 => 47
010100111 => 95
010101000 => 14
010101001 => 56
010101010 => 42
010101011 => 90
010101100 => 28
010101101 => 76
010101110 => 48
010101111 => 75
010110000 => 9
010110001 => 41
010110010 => 32
010110011 => 75
010110100 => 23
010110101 => 66
010110110 => 43
010110111 => 70
010111000 => 14
010111001 => 48
010111010 => 34
010111011 => 61
010111100 => 20
010111101 => 47
010111110 => 27
010111111 => 35
011000000 => 3
011000001 => 19
011000010 => 16
011000011 => 51
011000100 => 13
011000101 => 48
011000110 => 35
011000111 => 75
011001000 => 10
011001001 => 42
011001010 => 32
011001011 => 72
011001100 => 22
011001101 => 62
011001110 => 40
011001111 => 65
011010000 => 7
011010001 => 33
011010010 => 26
011010011 => 63
011010100 => 19
011010101 => 56
011010110 => 37
011010111 => 62
011011000 => 12
011011001 => 42
011011010 => 30
011011011 => 55
011011100 => 18
011011101 => 43
011011110 => 25
011011111 => 33
011100000 => 4
011100001 => 21
011100010 => 17
011100011 => 45
011100100 => 13
011100101 => 41
011100110 => 28
011100111 => 50
011101000 => 9
011101001 => 33
011101010 => 24
011101011 => 46
011101100 => 15
011101101 => 37
011101110 => 22
011101111 => 30
011110000 => 5
011110001 => 21
011110010 => 16
011110011 => 34
011110100 => 11
011110101 => 29
011110110 => 18
011110111 => 26
011111000 => 6
011111001 => 19
011111010 => 13
011111011 => 21
011111100 => 7
011111101 => 15
011111110 => 8
011111111 => 9
100000000 => 1
100000001 => 8
100000010 => 7
100000011 => 28
100000100 => 6
100000101 => 27
100000110 => 21
100000111 => 56
100001000 => 5
100001001 => 25
100001010 => 20
100001011 => 55
100001100 => 15
100001101 => 50
100001110 => 35
100001111 => 70
100010000 => 4
100010001 => 22
100010010 => 18
100010011 => 52
100010100 => 14
100010101 => 48
100010110 => 34
100010111 => 69
100011000 => 10
100011001 => 40
100011010 => 30
100011011 => 65
100011100 => 20
100011101 => 55
100011110 => 35
100011111 => 56
100100000 => 3
100100001 => 18
100100010 => 15
100100011 => 46
100100100 => 12
100100101 => 43
100100110 => 31
100100111 => 65
100101000 => 9
100101001 => 37
100101010 => 28
100101011 => 62
100101100 => 19
100101101 => 53
100101110 => 34
100101111 => 55
100110000 => 6
100110001 => 28
100110010 => 22
100110011 => 53
100110100 => 16
100110101 => 47
100110110 => 31
100110111 => 52
100111000 => 10
100111001 => 35
100111010 => 25
100111011 => 46
100111100 => 15
100111101 => 36
100111110 => 21
100111111 => 28
101000000 => 2
101000001 => 13
101000010 => 11
101000011 => 36
101000100 => 9
101000101 => 34
101000110 => 25
101000111 => 55
101001000 => 7
101001001 => 30
101001010 => 23
101001011 => 53
101001100 => 16
101001101 => 46
101001110 => 30
101001111 => 50
101010000 => 5
101010001 => 24
101010010 => 19
101010011 => 47
101010100 => 14
101010101 => 42
101010110 => 28
101010111 => 48
101011000 => 9
101011001 => 32
101011010 => 23
101011011 => 43
101011100 => 14
101011101 => 34
101011110 => 20
101011111 => 27
101100000 => 3
101100001 => 16
101100010 => 13
101100011 => 35
101100100 => 10
101100101 => 32
101100110 => 22
101100111 => 40
101101000 => 7
101101001 => 26
101101010 => 19
101101011 => 37
101101100 => 12
101101101 => 30
101101110 => 18
101101111 => 25
101110000 => 4
101110001 => 17
101110010 => 13
101110011 => 28
101110100 => 9
101110101 => 24
101110110 => 15
101110111 => 22
101111000 => 5
101111001 => 16
101111010 => 11
101111011 => 18
101111100 => 6
101111101 => 13
101111110 => 7
101111111 => 8
110000000 => 1
110000001 => 7
110000010 => 6
110000011 => 21
110000100 => 5
110000101 => 20
110000110 => 15
110000111 => 35
110001000 => 4
110001001 => 18
110001010 => 14
110001011 => 34
110001100 => 10
110001101 => 30
110001110 => 20
110001111 => 35
110010000 => 3
110010001 => 15
110010010 => 12
110010011 => 31
110010100 => 9
110010101 => 28
110010110 => 19
110010111 => 34
110011000 => 6
110011001 => 22
110011010 => 16
110011011 => 31
110011100 => 10
110011101 => 25
110011110 => 15
110011111 => 21
110100000 => 2
110100001 => 11
110100010 => 9
110100011 => 25
110100100 => 7
110100101 => 23
110100110 => 16
110100111 => 30
110101000 => 5
110101001 => 19
110101010 => 14
110101011 => 28
110101100 => 9
110101101 => 23
110101110 => 14
110101111 => 20
110110000 => 3
110110001 => 13
110110010 => 10
110110011 => 22
110110100 => 7
110110101 => 19
110110110 => 12
110110111 => 18
110111000 => 4
110111001 => 13
110111010 => 9
110111011 => 15
110111100 => 5
110111101 => 11
110111110 => 6
110111111 => 7
111000000 => 1
111000001 => 6
111000010 => 5
111000011 => 15
111000100 => 4
111000101 => 14
111000110 => 10
111000111 => 20
111001000 => 3
111001001 => 12
111001010 => 9
111001011 => 19
111001100 => 6
111001101 => 16
111001110 => 10
111001111 => 15
111010000 => 2
111010001 => 9
111010010 => 7
111010011 => 16
111010100 => 5
111010101 => 14
111010110 => 9
111010111 => 14
111011000 => 3
111011001 => 10
111011010 => 7
111011011 => 12
111011100 => 4
111011101 => 9
111011110 => 5
111011111 => 6
111100000 => 1
111100001 => 5
111100010 => 4
111100011 => 10
111100100 => 3
111100101 => 9
111100110 => 6
111100111 => 10
111101000 => 2
111101001 => 7
111101010 => 5
111101011 => 9
111101100 => 3
111101101 => 7
111101110 => 4
111101111 => 5
111110000 => 1
111110001 => 4
111110010 => 3
111110011 => 6
111110100 => 2
111110101 => 5
111110110 => 3
111110111 => 4
111111000 => 1
111111001 => 3
111111010 => 2
111111011 => 3
111111100 => 1
111111101 => 2
111111110 => 1
111111111 => 1
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Created: Dec 12, 2018 at 14:54 by Christian Stump
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Last Updated: Dec 12, 2018 at 14:54 by Christian Stump