edit this statistic or download as text // json
Identifier
Values
=>
0=>2 1=>1 00=>4 01=>3 10=>2 11=>1 000=>8 001=>7 010=>6 011=>4 100=>4 101=>3 110=>2 111=>1 0000=>16 0001=>15 0010=>14 0011=>11 0100=>12 0101=>10 0110=>8 0111=>5 1000=>8 1001=>7 1010=>6 1011=>4 1100=>4 1101=>3 1110=>2 1111=>1 00000=>32 00001=>31 00010=>30 00011=>26 00100=>28 00101=>25 00110=>22 00111=>16 01000=>24 01001=>22 01010=>20 01011=>15 01100=>16 01101=>13 01110=>10 01111=>6 10000=>16 10001=>15 10010=>14 10011=>11 10100=>12 10101=>10 10110=>8 10111=>5 11000=>8 11001=>7 11010=>6 11011=>4 11100=>4 11101=>3 11110=>2 11111=>1 000000=>64 000001=>63 000010=>62 000011=>57 000100=>60 000101=>56 000110=>52 000111=>42 001000=>56 001001=>53 001010=>50 001011=>41 001100=>44 001101=>38 001110=>32 001111=>22 010000=>48 010001=>46 010010=>44 010011=>37 010100=>40 010101=>35 010110=>30 010111=>21 011000=>32 011001=>29 011010=>26 011011=>19 011100=>20 011101=>16 011110=>12 011111=>7 100000=>32 100001=>31 100010=>30 100011=>26 100100=>28 100101=>25 100110=>22 100111=>16 101000=>24 101001=>22 101010=>20 101011=>15 101100=>16 101101=>13 101110=>10 101111=>6 110000=>16 110001=>15 110010=>14 110011=>11 110100=>12 110101=>10 110110=>8 110111=>5 111000=>8 111001=>7 111010=>6 111011=>4 111100=>4 111101=>3 111110=>2 111111=>1 0000000=>128 0000001=>127 0000010=>126 0000011=>120 0000100=>124 0000101=>119 0000110=>114 0000111=>99 0001000=>120 0001001=>116 0001010=>112 0001011=>98 0001100=>104 0001101=>94 0001110=>84 0001111=>64 0010000=>112 0010001=>109 0010010=>106 0010011=>94 0010100=>100 0010101=>91 0010110=>82 0010111=>63 0011000=>88 0011001=>82 0011010=>76 0011011=>60 0011100=>64 0011101=>54 0011110=>44 0011111=>29 0100000=>96 0100001=>94 0100010=>92 0100011=>83 0100100=>88 0100101=>81 0100110=>74 0100111=>58 0101000=>80 0101001=>75 0101010=>70 0101011=>56 0101100=>60 0101101=>51 0101110=>42 0101111=>28 0110000=>64 0110001=>61 0110010=>58 0110011=>48 0110100=>52 0110101=>45 0110110=>38 0110111=>26 0111000=>40 0111001=>36 0111010=>32 0111011=>23 0111100=>24 0111101=>19 0111110=>14 0111111=>8 1000000=>64 1000001=>63 1000010=>62 1000011=>57 1000100=>60 1000101=>56 1000110=>52 1000111=>42 1001000=>56 1001001=>53 1001010=>50 1001011=>41 1001100=>44 1001101=>38 1001110=>32 1001111=>22 1010000=>48 1010001=>46 1010010=>44 1010011=>37 1010100=>40 1010101=>35 1010110=>30 1010111=>21 1011000=>32 1011001=>29 1011010=>26 1011011=>19 1011100=>20 1011101=>16 1011110=>12 1011111=>7 1100000=>32 1100001=>31 1100010=>30 1100011=>26 1100100=>28 1100101=>25 1100110=>22 1100111=>16 1101000=>24 1101001=>22 1101010=>20 1101011=>15 1101100=>16 1101101=>13 1101110=>10 1101111=>6 1110000=>16 1110001=>15 1110010=>14 1110011=>11 1110100=>12 1110101=>10 1110110=>8 1110111=>5 1111000=>8 1111001=>7 1111010=>6 1111011=>4 1111100=>4 1111101=>3 1111110=>2 1111111=>1 00000000=>256 00000001=>255 00000010=>254 00000011=>247 00000100=>252 00000101=>246 00000110=>240 00000111=>219 00001000=>248 00001001=>243 00001010=>238 00001011=>218 00001100=>228 00001101=>213 00001110=>198 00001111=>163 00010000=>240 00010001=>236 00010010=>232 00010011=>214 00010100=>224 00010101=>210 00010110=>196 00010111=>162 00011000=>208 00011001=>198 00011010=>188 00011011=>158 00011100=>168 00011101=>148 00011110=>128 00011111=>93 00100000=>224 00100001=>221 00100010=>218 00100011=>203 00100100=>212 00100101=>200 00100110=>188 00100111=>157 00101000=>200 00101001=>191 00101010=>182 00101011=>154 00101100=>164 00101101=>145 00101110=>126 00101111=>92 00110000=>176 00110001=>170 00110010=>164 00110011=>142 00110100=>152 00110101=>136 00110110=>120 00110111=>89 00111000=>128 00111001=>118 00111010=>108 00111011=>83 00111100=>88 00111101=>73 00111110=>58 00111111=>37 01000000=>192 01000001=>190 01000010=>188 01000011=>177 01000100=>184 01000101=>175 01000110=>166 01000111=>141 01001000=>176 01001001=>169 01001010=>162 01001011=>139 01001100=>148 01001101=>132 01001110=>116 01001111=>86 01010000=>160 01010001=>155 01010010=>150 01010011=>131 01010100=>140 01010101=>126 01010110=>112 01010111=>84 01011000=>120 01011001=>111 01011010=>102 01011011=>79 01011100=>84 01011101=>70 01011110=>56 01011111=>36 01100000=>128 01100001=>125 01100010=>122 01100011=>109 01100100=>116 01100101=>106 01100110=>96 01100111=>74 01101000=>104 01101001=>97 01101010=>90 01101011=>71 01101100=>76 01101101=>64 01101110=>52 01101111=>34 01110000=>80 01110001=>76 01110010=>72 01110011=>59 01110100=>64 01110101=>55 01110110=>46 01110111=>31 01111000=>48 01111001=>43 01111010=>38 01111011=>27 01111100=>28 01111101=>22 01111110=>16 01111111=>9 10000000=>128 10000001=>127 10000010=>126 10000011=>120 10000100=>124 10000101=>119 10000110=>114 10000111=>99 10001000=>120 10001001=>116 10001010=>112 10001011=>98 10001100=>104 10001101=>94 10001110=>84 10001111=>64 10010000=>112 10010001=>109 10010010=>106 10010011=>94 10010100=>100 10010101=>91 10010110=>82 10010111=>63 10011000=>88 10011001=>82 10011010=>76 10011011=>60 10011100=>64 10011101=>54 10011110=>44 10011111=>29 10100000=>96 10100001=>94 10100010=>92 10100011=>83 10100100=>88 10100101=>81 10100110=>74 10100111=>58 10101000=>80 10101001=>75 10101010=>70 10101011=>56 10101100=>60 10101101=>51 10101110=>42 10101111=>28 10110000=>64 10110001=>61 10110010=>58 10110011=>48 10110100=>52 10110101=>45 10110110=>38 10110111=>26 10111000=>40 10111001=>36 10111010=>32 10111011=>23 10111100=>24 10111101=>19 10111110=>14 10111111=>8 11000000=>64 11000001=>63 11000010=>62 11000011=>57 11000100=>60 11000101=>56 11000110=>52 11000111=>42 11001000=>56 11001001=>53 11001010=>50 11001011=>41 11001100=>44 11001101=>38 11001110=>32 11001111=>22 11010000=>48 11010001=>46 11010010=>44 11010011=>37 11010100=>40 11010101=>35 11010110=>30 11010111=>21 11011000=>32 11011001=>29 11011010=>26 11011011=>19 11011100=>20 11011101=>16 11011110=>12 11011111=>7 11100000=>32 11100001=>31 11100010=>30 11100011=>26 11100100=>28 11100101=>25 11100110=>22 11100111=>16 11101000=>24 11101001=>22 11101010=>20 11101011=>15 11101100=>16 11101101=>13 11101110=>10 11101111=>6 11110000=>16 11110001=>15 11110010=>14 11110011=>11 11110100=>12 11110101=>10 11110110=>8 11110111=>5 11111000=>8 11111001=>7 11111010=>6 11111011=>4 11111100=>4 11111101=>3 11111110=>2 11111111=>1 000000000=>512 000000001=>511 000000010=>510 000000011=>502 000000100=>508 000000101=>501 000000110=>494 000000111=>466 000001000=>504 000001001=>498 000001010=>492 000001011=>465 000001100=>480 000001101=>459 000001110=>438 000001111=>382 000010000=>496 000010001=>491 000010010=>486 000010011=>461 000010100=>476 000010101=>456 000010110=>436 000010111=>381 000011000=>456 000011001=>441 000011010=>426 000011011=>376 000011100=>396 000011101=>361 000011110=>326 000011111=>256 000100000=>480 000100001=>476 000100010=>472 000100011=>450 000100100=>464 000100101=>446 000100110=>428 000100111=>376 000101000=>448 000101001=>434 000101010=>420 000101011=>372 000101100=>392 000101101=>358 000101110=>324 000101111=>255 000110000=>416 000110001=>406 000110010=>396 000110011=>356 000110100=>376 000110101=>346 000110110=>316 000110111=>251 000111000=>336 000111001=>316 000111010=>296 000111011=>241 000111100=>256 000111101=>221 000111110=>186 000111111=>130 001000000=>448 001000001=>445 001000010=>442 001000011=>424 001000100=>436 001000101=>421 001000110=>406 001000111=>360 001001000=>424 001001001=>412 001001010=>400 001001011=>357 001001100=>376 001001101=>345 001001110=>314 001001111=>249 001010000=>400 001010001=>391 001010010=>382 001010011=>345 001010100=>364 001010101=>336 001010110=>308 001010111=>246 001011000=>328 001011001=>309 001011010=>290 001011011=>237 001011100=>252 001011101=>218 001011110=>184 001011111=>129 001100000=>352 001100001=>346 001100010=>340 001100011=>312 001100100=>328 001100101=>306 001100110=>284 001100111=>231 001101000=>304 001101001=>288 001101010=>272 001101011=>225 001101100=>240 001101101=>209 001101110=>178 001101111=>126 001110000=>256 001110001=>246 001110010=>236 001110011=>201 001110100=>216 001110101=>191 001110110=>166 001110111=>120 001111000=>176 001111001=>161 001111010=>146 001111011=>110 001111100=>116 001111101=>95 001111110=>74 001111111=>46 010000000=>384 010000001=>382 010000010=>380 010000011=>367 010000100=>376 010000101=>365 010000110=>354 010000111=>318 010001000=>368 010001001=>359 010001010=>350 010001011=>316 010001100=>332 010001101=>307 010001110=>282 010001111=>227 010010000=>352 010010001=>345 010010010=>338 010010011=>308 010010100=>324 010010101=>301 010010110=>278 010010111=>225 010011000=>296 010011001=>280 010011010=>264 010011011=>218 010011100=>232 010011101=>202 010011110=>172 010011111=>122 010100000=>320 010100001=>315 010100010=>310 010100011=>286 010100100=>300 010100101=>281 010100110=>262 010100111=>215 010101000=>280 010101001=>266 010101010=>252 010101011=>210 010101100=>224 010101101=>196 010101110=>168 010101111=>120 010110000=>240 010110001=>231 010110010=>222 010110011=>190 010110100=>204 010110101=>181 010110110=>158 010110111=>115 010111000=>168 010111001=>154 010111010=>140 010111011=>106 010111100=>112 010111101=>92 010111110=>72 010111111=>45 011000000=>256 011000001=>253 011000010=>250 011000011=>234 011000100=>244 011000101=>231 011000110=>218 011000111=>183 011001000=>232 011001001=>222 011001010=>212 011001011=>180 011001100=>192 011001101=>170 011001110=>148 011001111=>108 011010000=>208 011010001=>201 011010010=>194 011010011=>168 011010100=>180 011010101=>161 011010110=>142 011010111=>105 011011000=>152 011011001=>140 011011010=>128 011011011=>98 011011100=>104 011011101=>86 011011110=>68 011011111=>43 011100000=>160 011100001=>156 011100010=>152 011100011=>135 011100100=>144 011100101=>131 011100110=>118 011100111=>90 011101000=>128 011101001=>119 011101010=>110 011101011=>86 011101100=>92 011101101=>77 011101110=>62 011101111=>40 011110000=>96 011110001=>91 011110010=>86 011110011=>70 011110100=>76 011110101=>65 011110110=>54 011110111=>36 011111000=>56 011111001=>50 011111010=>44 011111011=>31 011111100=>32 011111101=>25 011111110=>18 011111111=>10 100000000=>256 100000001=>255 100000010=>254 100000011=>247 100000100=>252 100000101=>246 100000110=>240 100000111=>219 100001000=>248 100001001=>243 100001010=>238 100001011=>218 100001100=>228 100001101=>213 100001110=>198 100001111=>163 100010000=>240 100010001=>236 100010010=>232 100010011=>214 100010100=>224 100010101=>210 100010110=>196 100010111=>162 100011000=>208 100011001=>198 100011010=>188 100011011=>158 100011100=>168 100011101=>148 100011110=>128 100011111=>93 100100000=>224 100100001=>221 100100010=>218 100100011=>203 100100100=>212 100100101=>200 100100110=>188 100100111=>157 100101000=>200 100101001=>191 100101010=>182 100101011=>154 100101100=>164 100101101=>145 100101110=>126 100101111=>92 100110000=>176 100110001=>170 100110010=>164 100110011=>142 100110100=>152 100110101=>136 100110110=>120 100110111=>89 100111000=>128 100111001=>118 100111010=>108 100111011=>83 100111100=>88 100111101=>73 100111110=>58 100111111=>37 101000000=>192 101000001=>190 101000010=>188 101000011=>177 101000100=>184 101000101=>175 101000110=>166 101000111=>141 101001000=>176 101001001=>169 101001010=>162 101001011=>139 101001100=>148 101001101=>132 101001110=>116 101001111=>86 101010000=>160 101010001=>155 101010010=>150 101010011=>131 101010100=>140 101010101=>126 101010110=>112 101010111=>84 101011000=>120 101011001=>111 101011010=>102 101011011=>79 101011100=>84 101011101=>70 101011110=>56 101011111=>36 101100000=>128 101100001=>125 101100010=>122 101100011=>109 101100100=>116 101100101=>106 101100110=>96 101100111=>74 101101000=>104 101101001=>97 101101010=>90 101101011=>71 101101100=>76 101101101=>64 101101110=>52 101101111=>34 101110000=>80 101110001=>76 101110010=>72 101110011=>59 101110100=>64 101110101=>55 101110110=>46 101110111=>31 101111000=>48 101111001=>43 101111010=>38 101111011=>27 101111100=>28 101111101=>22 101111110=>16 101111111=>9 110000000=>128 110000001=>127 110000010=>126 110000011=>120 110000100=>124 110000101=>119 110000110=>114 110000111=>99 110001000=>120 110001001=>116 110001010=>112 110001011=>98 110001100=>104 110001101=>94 110001110=>84 110001111=>64 110010000=>112 110010001=>109 110010010=>106 110010011=>94 110010100=>100 110010101=>91 110010110=>82 110010111=>63 110011000=>88 110011001=>82 110011010=>76 110011011=>60 110011100=>64 110011101=>54 110011110=>44 110011111=>29 110100000=>96 110100001=>94 110100010=>92 110100011=>83 110100100=>88 110100101=>81 110100110=>74 110100111=>58 110101000=>80 110101001=>75 110101010=>70 110101011=>56 110101100=>60 110101101=>51 110101110=>42 110101111=>28 110110000=>64 110110001=>61 110110010=>58 110110011=>48 110110100=>52 110110101=>45 110110110=>38 110110111=>26 110111000=>40 110111001=>36 110111010=>32 110111011=>23 110111100=>24 110111101=>19 110111110=>14 110111111=>8 111000000=>64 111000001=>63 111000010=>62 111000011=>57 111000100=>60 111000101=>56 111000110=>52 111000111=>42 111001000=>56 111001001=>53 111001010=>50 111001011=>41 111001100=>44 111001101=>38 111001110=>32 111001111=>22 111010000=>48 111010001=>46 111010010=>44 111010011=>37 111010100=>40 111010101=>35 111010110=>30 111010111=>21 111011000=>32 111011001=>29 111011010=>26 111011011=>19 111011100=>20 111011101=>16 111011110=>12 111011111=>7 111100000=>32 111100001=>31 111100010=>30 111100011=>26 111100100=>28 111100101=>25 111100110=>22 111100111=>16 111101000=>24 111101001=>22 111101010=>20 111101011=>15 111101100=>16 111101101=>13 111101110=>10 111101111=>6 111110000=>16 111110001=>15 111110010=>14 111110011=>11 111110100=>12 111110101=>10 111110110=>8 111110111=>5 111111000=>8 111111001=>7 111111010=>6 111111011=>4 111111100=>4 111111101=>3 111111110=>2 111111111=>1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The number of lattice paths of the same length weakly above the path given by a binary word.
In particular, there are $2^n$ lattice paths weakly above the the length $n$ binary word $0\dots 0$, there is a unique path weakly above $1\dots 1$, and there are $\binom{2n}{n}$ paths weakly above the length $2n$ binary word $10\dots 10$.
Code
@cached_function
def all_paths_above(L, i=0):
    if i < 0:
        return []
    elif len(L) == 0:
        return [tuple()]

    else:
        steps = []
        if L[0] == 1:
            steps.append((1,i))
            steps.append((0,i-1))
        if L[0] == 0:
            steps.append((1,i+1))
            steps.append((0,i))
        return [ tuple([a]) + path for a,i in steps for path in all_paths_above(L[1:],i)]

def statistic(D):
    return len(all_paths_above(tuple(D)))
Created
Mar 16, 2019 at 14:35 by Martin Rubey
Updated
Mar 16, 2019 at 14:35 by Martin Rubey