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Statistic identifier: St001915
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Collection: Binary words
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Description: The size of the component corresponding to a necklace in Bulgarian solitaire.
A move in Bulgarian solitaire consists of removing the first column of the Ferrers diagram and inserting it as a new row.
The connected components of the corresponding discrete dynamical system are indexed by necklaces in a natural way.
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References: [1] (service unreachable) [[MathSciNet:0656129]]
[2] Society, pages 483–486, 1982
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Code:
def move(la):
return Partition(sorted([p-1 for p in la] + [len(la)], reverse=True))
def necklace_to_partition(w):
m = len(w)
la = [m - i + b for i, b in enumerate(w, 1)]
return Partition(la)
@cached_function
def components(n):
B = DiscreteDynamicalSystem(Partitions(n), move)
C = {frozenset(c): set(c) for c in B.cycles()}
E = set(B).difference(*C.values())
while E:
e = E.pop()
if not any(e in c for c in C.values()):
o, k = B.orbit(e, True)
c = frozenset(o[k:])
r = o[:k]
C[c].update(r)
E.difference_update(r)
return C
def component(la):
la = Partition(la)
n = la.size()
B = DiscreteDynamicalSystem(Partitions(n), move)
o, k = B.orbit(la, preperiod=True)
return components(n)[frozenset(o[k:])]
def statistic(w):
return len(component(necklace_to_partition(w)))
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Statistic values:
0 => 1
1 => 1
00 => 1
01 => 2
10 => 2
11 => 3
000 => 3
001 => 5
010 => 5
011 => 7
100 => 5
101 => 7
110 => 7
111 => 11
0000 => 11
0001 => 15
0010 => 15
0011 => 15
0100 => 15
0101 => 7
0110 => 15
0111 => 30
1000 => 15
1001 => 15
1010 => 7
1011 => 30
1100 => 15
1101 => 30
1110 => 30
1111 => 42
00000 => 42
00001 => 56
00010 => 56
00011 => 45
00100 => 56
00101 => 32
00110 => 45
00111 => 67
01000 => 56
01001 => 32
01010 => 32
01011 => 34
01100 => 45
01101 => 34
01110 => 67
01111 => 135
10000 => 56
10001 => 45
10010 => 32
10011 => 67
10100 => 32
10101 => 34
10110 => 34
10111 => 135
11000 => 45
11001 => 67
11010 => 34
11011 => 135
11100 => 67
11101 => 135
11110 => 135
11111 => 176
000000 => 176
000001 => 231
000010 => 231
000011 => 185
000100 => 231
000101 => 87
000110 => 185
000111 => 214
001000 => 231
001001 => 25
001010 => 87
001011 => 80
001100 => 185
001101 => 65
001110 => 214
001111 => 322
010000 => 231
010001 => 87
010010 => 25
010011 => 65
010100 => 87
010101 => 26
010110 => 80
010111 => 133
011000 => 185
011001 => 80
011010 => 65
011011 => 35
011100 => 214
011101 => 133
011110 => 322
011111 => 627
100000 => 231
100001 => 185
100010 => 87
100011 => 214
100100 => 25
100101 => 80
100110 => 65
100111 => 322
101000 => 87
101001 => 65
101010 => 26
101011 => 133
101100 => 80
101101 => 35
101110 => 133
101111 => 627
110000 => 185
110001 => 214
110010 => 80
110011 => 322
110100 => 65
110101 => 133
110110 => 35
110111 => 627
111000 => 214
111001 => 322
111010 => 133
111011 => 627
111100 => 322
111101 => 627
111110 => 627
111111 => 792
0000000 => 792
0000001 => 1002
0000010 => 1002
0000011 => 811
0000100 => 1002
0000101 => 294
0000110 => 811
0000111 => 777
0001000 => 1002
0001001 => 150
0001010 => 294
0001011 => 336
0001100 => 811
0001101 => 189
0001110 => 777
0001111 => 1114
0010000 => 1002
0010001 => 150
0010010 => 150
0010011 => 125
0010100 => 294
0010101 => 148
0010110 => 336
0010111 => 286
0011000 => 811
0011001 => 125
0011010 => 189
0011011 => 145
0011100 => 777
0011101 => 255
0011110 => 1114
0011111 => 1637
0100000 => 1002
0100001 => 294
0100010 => 150
0100011 => 189
0100100 => 150
0100101 => 148
0100110 => 125
0100111 => 255
0101000 => 294
0101001 => 148
0101010 => 148
0101011 => 158
0101100 => 336
0101101 => 158
0101110 => 286
0101111 => 544
0110000 => 811
0110001 => 336
0110010 => 125
0110011 => 145
0110100 => 189
0110101 => 158
0110110 => 145
0110111 => 255
0111000 => 777
0111001 => 286
0111010 => 255
0111011 => 255
0111100 => 1114
0111101 => 544
0111110 => 1637
0111111 => 3010
1000000 => 1002
1000001 => 811
1000010 => 294
1000011 => 777
1000100 => 150
1000101 => 336
1000110 => 189
1000111 => 1114
1001000 => 150
1001001 => 125
1001010 => 148
1001011 => 286
1001100 => 125
1001101 => 145
1001110 => 255
1001111 => 1637
1010000 => 294
1010001 => 189
1010010 => 148
1010011 => 255
1010100 => 148
1010101 => 158
1010110 => 158
1010111 => 544
1011000 => 336
1011001 => 145
1011010 => 158
1011011 => 255
1011100 => 286
1011101 => 255
1011110 => 544
1011111 => 3010
1100000 => 811
1100001 => 777
1100010 => 336
1100011 => 1114
1100100 => 125
1100101 => 286
1100110 => 145
1100111 => 1637
1101000 => 189
1101001 => 255
1101010 => 158
1101011 => 544
1101100 => 145
1101101 => 255
1101110 => 255
1101111 => 3010
1110000 => 777
1110001 => 1114
1110010 => 286
1110011 => 1637
1110100 => 255
1110101 => 544
1110110 => 255
1110111 => 3010
1111000 => 1114
1111001 => 1637
1111010 => 544
1111011 => 3010
1111100 => 1637
1111101 => 3010
1111110 => 3010
1111111 => 3718
00000000 => 3718
00000001 => 4565
00000010 => 4565
00000011 => 3727
00000100 => 4565
00000101 => 1152
00000110 => 3727
00000111 => 3880
00001000 => 4565
00001001 => 500
00001010 => 1152
00001011 => 815
00001100 => 3727
00001101 => 747
00001110 => 3880
00001111 => 4420
00010000 => 4565
00010001 => 225
00010010 => 500
00010011 => 375
00010100 => 1152
00010101 => 420
00010110 => 815
00010111 => 1076
00011000 => 3727
00011001 => 360
00011010 => 747
00011011 => 414
00011100 => 3880
00011101 => 802
00011110 => 4420
00011111 => 5972
00100000 => 4565
00100001 => 500
00100010 => 225
00100011 => 360
00100100 => 500
00100101 => 245
00100110 => 375
00100111 => 470
00101000 => 1152
00101001 => 245
00101010 => 420
00101011 => 309
00101100 => 815
00101101 => 316
00101110 => 1076
00101111 => 1082
00110000 => 3727
00110001 => 375
00110010 => 360
00110011 => 150
00110100 => 747
00110101 => 295
00110110 => 414
00110111 => 525
00111000 => 3880
00111001 => 470
00111010 => 802
00111011 => 524
00111100 => 4420
00111101 => 1158
00111110 => 5972
00111111 => 8463
01000000 => 4565
01000001 => 1152
01000010 => 500
01000011 => 747
01000100 => 225
01000101 => 420
01000110 => 360
01000111 => 802
01001000 => 500
01001001 => 245
01001010 => 245
01001011 => 316
01001100 => 375
01001101 => 295
01001110 => 470
01001111 => 1158
01010000 => 1152
01010001 => 420
01010010 => 245
01010011 => 295
01010100 => 420
01010101 => 97
01010110 => 309
01010111 => 593
01011000 => 815
01011001 => 309
01011010 => 316
01011011 => 289
01011100 => 1076
01011101 => 593
01011110 => 1082
01011111 => 2414
01100000 => 3727
01100001 => 815
01100010 => 375
01100011 => 414
01100100 => 360
01100101 => 309
01100110 => 150
01100111 => 524
01101000 => 747
01101001 => 316
01101010 => 295
01101011 => 289
01101100 => 414
01101101 => 289
01101110 => 525
01101111 => 983
01110000 => 3880
01110001 => 1076
01110010 => 470
01110011 => 525
01110100 => 802
01110101 => 593
01110110 => 524
01110111 => 450
01111000 => 4420
01111001 => 1082
01111010 => 1158
01111011 => 983
01111100 => 5972
01111101 => 2414
01111110 => 8463
01111111 => 14883
10000000 => 4565
10000001 => 3727
10000010 => 1152
10000011 => 3880
10000100 => 500
10000101 => 815
10000110 => 747
10000111 => 4420
10001000 => 225
10001001 => 375
10001010 => 420
10001011 => 1076
10001100 => 360
10001101 => 414
10001110 => 802
10001111 => 5972
10010000 => 500
10010001 => 360
10010010 => 245
10010011 => 470
10010100 => 245
10010101 => 309
10010110 => 316
10010111 => 1082
10011000 => 375
10011001 => 150
10011010 => 295
10011011 => 525
10011100 => 470
10011101 => 524
10011110 => 1158
10011111 => 8463
10100000 => 1152
10100001 => 747
10100010 => 420
10100011 => 802
10100100 => 245
10100101 => 316
10100110 => 295
10100111 => 1158
10101000 => 420
10101001 => 295
10101010 => 97
10101011 => 593
10101100 => 309
10101101 => 289
10101110 => 593
10101111 => 2414
10110000 => 815
10110001 => 414
10110010 => 309
10110011 => 524
10110100 => 316
10110101 => 289
10110110 => 289
10110111 => 983
10111000 => 1076
10111001 => 525
10111010 => 593
10111011 => 450
10111100 => 1082
10111101 => 983
10111110 => 2414
10111111 => 14883
11000000 => 3727
11000001 => 3880
11000010 => 815
11000011 => 4420
11000100 => 375
11000101 => 1076
11000110 => 414
11000111 => 5972
11001000 => 360
11001001 => 470
11001010 => 309
11001011 => 1082
11001100 => 150
11001101 => 525
11001110 => 524
11001111 => 8463
11010000 => 747
11010001 => 802
11010010 => 316
11010011 => 1158
11010100 => 295
11010101 => 593
11010110 => 289
11010111 => 2414
11011000 => 414
11011001 => 524
11011010 => 289
11011011 => 983
11011100 => 525
11011101 => 450
11011110 => 983
11011111 => 14883
11100000 => 3880
11100001 => 4420
11100010 => 1076
11100011 => 5972
11100100 => 470
11100101 => 1082
11100110 => 525
11100111 => 8463
11101000 => 802
11101001 => 1158
11101010 => 593
11101011 => 2414
11101100 => 524
11101101 => 983
11101110 => 450
11101111 => 14883
11110000 => 4420
11110001 => 5972
11110010 => 1082
11110011 => 8463
11110100 => 1158
11110101 => 2414
11110110 => 983
11110111 => 14883
11111000 => 5972
11111001 => 8463
11111010 => 2414
11111011 => 14883
11111100 => 8463
11111101 => 14883
11111110 => 14883
11111111 => 17977
000000000 => 17977
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Created: Aug 16, 2023 at 08:53 by Martin Rubey
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Last Updated: Aug 16, 2023 at 08:53 by Martin Rubey