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Identifier
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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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 St001312Number of parabolic noncrossing partitions indexed by the composition. for this statistic on compositions treated as bounce paths.
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))
Created
Dec 12, 2018 at 14:54 by Christian Stump
Updated
Dec 12, 2018 at 14:54 by Christian Stump