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Statistic identifier: St001916
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Collection: Binary words
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Description: The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace.
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. The binary words corresponding to a necklace index the recurrent elements in a component, the remaining elements are called transient.
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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)
def orbit(la):
la = Partition(la)
n = la.size()
B = DiscreteDynamicalSystem(Partitions(n), move)
return B.orbit(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))) - len(w.conjugates())
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Statistic values:
0 => 0
1 => 0
00 => 0
01 => 0
10 => 0
11 => 2
000 => 2
001 => 2
010 => 2
011 => 4
100 => 2
101 => 4
110 => 4
111 => 10
0000 => 10
0001 => 11
0010 => 11
0011 => 11
0100 => 11
0101 => 5
0110 => 11
0111 => 26
1000 => 11
1001 => 11
1010 => 5
1011 => 26
1100 => 11
1101 => 26
1110 => 26
1111 => 41
00000 => 41
00001 => 51
00010 => 51
00011 => 40
00100 => 51
00101 => 27
00110 => 40
00111 => 62
01000 => 51
01001 => 27
01010 => 27
01011 => 29
01100 => 40
01101 => 29
01110 => 62
01111 => 130
10000 => 51
10001 => 40
10010 => 27
10011 => 62
10100 => 27
10101 => 29
10110 => 29
10111 => 130
11000 => 40
11001 => 62
11010 => 29
11011 => 130
11100 => 62
11101 => 130
11110 => 130
11111 => 175
000000 => 175
000001 => 225
000010 => 225
000011 => 179
000100 => 225
000101 => 81
000110 => 179
000111 => 208
001000 => 225
001001 => 22
001010 => 81
001011 => 74
001100 => 179
001101 => 59
001110 => 208
001111 => 316
010000 => 225
010001 => 81
010010 => 22
010011 => 59
010100 => 81
010101 => 24
010110 => 74
010111 => 127
011000 => 179
011001 => 74
011010 => 59
011011 => 32
011100 => 208
011101 => 127
011110 => 316
011111 => 621
100000 => 225
100001 => 179
100010 => 81
100011 => 208
100100 => 22
100101 => 74
100110 => 59
100111 => 316
101000 => 81
101001 => 59
101010 => 24
101011 => 127
101100 => 74
101101 => 32
101110 => 127
101111 => 621
110000 => 179
110001 => 208
110010 => 74
110011 => 316
110100 => 59
110101 => 127
110110 => 32
110111 => 621
111000 => 208
111001 => 316
111010 => 127
111011 => 621
111100 => 316
111101 => 621
111110 => 621
111111 => 791
0000000 => 791
0000001 => 995
0000010 => 995
0000011 => 804
0000100 => 995
0000101 => 287
0000110 => 804
0000111 => 770
0001000 => 995
0001001 => 143
0001010 => 287
0001011 => 329
0001100 => 804
0001101 => 182
0001110 => 770
0001111 => 1107
0010000 => 995
0010001 => 143
0010010 => 143
0010011 => 118
0010100 => 287
0010101 => 141
0010110 => 329
0010111 => 279
0011000 => 804
0011001 => 118
0011010 => 182
0011011 => 138
0011100 => 770
0011101 => 248
0011110 => 1107
0011111 => 1630
0100000 => 995
0100001 => 287
0100010 => 143
0100011 => 182
0100100 => 143
0100101 => 141
0100110 => 118
0100111 => 248
0101000 => 287
0101001 => 141
0101010 => 141
0101011 => 151
0101100 => 329
0101101 => 151
0101110 => 279
0101111 => 537
0110000 => 804
0110001 => 329
0110010 => 118
0110011 => 138
0110100 => 182
0110101 => 151
0110110 => 138
0110111 => 248
0111000 => 770
0111001 => 279
0111010 => 248
0111011 => 248
0111100 => 1107
0111101 => 537
0111110 => 1630
0111111 => 3003
1000000 => 995
1000001 => 804
1000010 => 287
1000011 => 770
1000100 => 143
1000101 => 329
1000110 => 182
1000111 => 1107
1001000 => 143
1001001 => 118
1001010 => 141
1001011 => 279
1001100 => 118
1001101 => 138
1001110 => 248
1001111 => 1630
1010000 => 287
1010001 => 182
1010010 => 141
1010011 => 248
1010100 => 141
1010101 => 151
1010110 => 151
1010111 => 537
1011000 => 329
1011001 => 138
1011010 => 151
1011011 => 248
1011100 => 279
1011101 => 248
1011110 => 537
1011111 => 3003
1100000 => 804
1100001 => 770
1100010 => 329
1100011 => 1107
1100100 => 118
1100101 => 279
1100110 => 138
1100111 => 1630
1101000 => 182
1101001 => 248
1101010 => 151
1101011 => 537
1101100 => 138
1101101 => 248
1101110 => 248
1101111 => 3003
1110000 => 770
1110001 => 1107
1110010 => 279
1110011 => 1630
1110100 => 248
1110101 => 537
1110110 => 248
1110111 => 3003
1111000 => 1107
1111001 => 1630
1111010 => 537
1111011 => 3003
1111100 => 1630
1111101 => 3003
1111110 => 3003
1111111 => 3717
00000000 => 3717
00000001 => 4557
00000010 => 4557
00000011 => 3719
00000100 => 4557
00000101 => 1144
00000110 => 3719
00000111 => 3872
00001000 => 4557
00001001 => 492
00001010 => 1144
00001011 => 807
00001100 => 3719
00001101 => 739
00001110 => 3872
00001111 => 4412
00010000 => 4557
00010001 => 221
00010010 => 492
00010011 => 367
00010100 => 1144
00010101 => 412
00010110 => 807
00010111 => 1068
00011000 => 3719
00011001 => 352
00011010 => 739
00011011 => 406
00011100 => 3872
00011101 => 794
00011110 => 4412
00011111 => 5964
00100000 => 4557
00100001 => 492
00100010 => 221
00100011 => 352
00100100 => 492
00100101 => 237
00100110 => 367
00100111 => 462
00101000 => 1144
00101001 => 237
00101010 => 412
00101011 => 301
00101100 => 807
00101101 => 308
00101110 => 1068
00101111 => 1074
00110000 => 3719
00110001 => 367
00110010 => 352
00110011 => 146
00110100 => 739
00110101 => 287
00110110 => 406
00110111 => 517
00111000 => 3872
00111001 => 462
00111010 => 794
00111011 => 516
00111100 => 4412
00111101 => 1150
00111110 => 5964
00111111 => 8455
01000000 => 4557
01000001 => 1144
01000010 => 492
01000011 => 739
01000100 => 221
01000101 => 412
01000110 => 352
01000111 => 794
01001000 => 492
01001001 => 237
01001010 => 237
01001011 => 308
01001100 => 367
01001101 => 287
01001110 => 462
01001111 => 1150
01010000 => 1144
01010001 => 412
01010010 => 237
01010011 => 287
01010100 => 412
01010101 => 95
01010110 => 301
01010111 => 585
01011000 => 807
01011001 => 301
01011010 => 308
01011011 => 281
01011100 => 1068
01011101 => 585
01011110 => 1074
01011111 => 2406
01100000 => 3719
01100001 => 807
01100010 => 367
01100011 => 406
01100100 => 352
01100101 => 301
01100110 => 146
01100111 => 516
01101000 => 739
01101001 => 308
01101010 => 287
01101011 => 281
01101100 => 406
01101101 => 281
01101110 => 517
01101111 => 975
01110000 => 3872
01110001 => 1068
01110010 => 462
01110011 => 517
01110100 => 794
01110101 => 585
01110110 => 516
01110111 => 446
01111000 => 4412
01111001 => 1074
01111010 => 1150
01111011 => 975
01111100 => 5964
01111101 => 2406
01111110 => 8455
01111111 => 14875
10000000 => 4557
10000001 => 3719
10000010 => 1144
10000011 => 3872
10000100 => 492
10000101 => 807
10000110 => 739
10000111 => 4412
10001000 => 221
10001001 => 367
10001010 => 412
10001011 => 1068
10001100 => 352
10001101 => 406
10001110 => 794
10001111 => 5964
10010000 => 492
10010001 => 352
10010010 => 237
10010011 => 462
10010100 => 237
10010101 => 301
10010110 => 308
10010111 => 1074
10011000 => 367
10011001 => 146
10011010 => 287
10011011 => 517
10011100 => 462
10011101 => 516
10011110 => 1150
10011111 => 8455
10100000 => 1144
10100001 => 739
10100010 => 412
10100011 => 794
10100100 => 237
10100101 => 308
10100110 => 287
10100111 => 1150
10101000 => 412
10101001 => 287
10101010 => 95
10101011 => 585
10101100 => 301
10101101 => 281
10101110 => 585
10101111 => 2406
10110000 => 807
10110001 => 406
10110010 => 301
10110011 => 516
10110100 => 308
10110101 => 281
10110110 => 281
10110111 => 975
10111000 => 1068
10111001 => 517
10111010 => 585
10111011 => 446
10111100 => 1074
10111101 => 975
10111110 => 2406
10111111 => 14875
11000000 => 3719
11000001 => 3872
11000010 => 807
11000011 => 4412
11000100 => 367
11000101 => 1068
11000110 => 406
11000111 => 5964
11001000 => 352
11001001 => 462
11001010 => 301
11001011 => 1074
11001100 => 146
11001101 => 517
11001110 => 516
11001111 => 8455
11010000 => 739
11010001 => 794
11010010 => 308
11010011 => 1150
11010100 => 287
11010101 => 585
11010110 => 281
11010111 => 2406
11011000 => 406
11011001 => 516
11011010 => 281
11011011 => 975
11011100 => 517
11011101 => 446
11011110 => 975
11011111 => 14875
11100000 => 3872
11100001 => 4412
11100010 => 1068
11100011 => 5964
11100100 => 462
11100101 => 1074
11100110 => 517
11100111 => 8455
11101000 => 794
11101001 => 1150
11101010 => 585
11101011 => 2406
11101100 => 516
11101101 => 975
11101110 => 446
11101111 => 14875
11110000 => 4412
11110001 => 5964
11110010 => 1074
11110011 => 8455
11110100 => 1150
11110101 => 2406
11110110 => 975
11110111 => 14875
11111000 => 5964
11111001 => 8455
11111010 => 2406
11111011 => 14875
11111100 => 8455
11111101 => 14875
11111110 => 14875
11111111 => 17976
000000000 => 17976
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Created: Aug 16, 2023 at 09:15 by Martin Rubey
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Last Updated: Aug 16, 2023 at 09:15 by Martin Rubey