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* www.FindStat.org - The Combinatorial Statistic Finder *
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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *
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-----------------------------------------------------------------------------
Statistic identifier: St001365
-----------------------------------------------------------------------------
Collection: Binary words
-----------------------------------------------------------------------------
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 $10\dots 10$.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
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)))
-----------------------------------------------------------------------------
Statistic 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
-----------------------------------------------------------------------------
Created: Mar 16, 2019 at 14:35 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Mar 16, 2019 at 14:35 by Martin Rubey