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Identifier
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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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.
References
[1] (service unreachable) MathSciNet:0656129
[2] Society, pages 483–486, 1982
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())
Created
Aug 16, 2023 at 09:15 by Martin Rubey
Updated
Aug 16, 2023 at 09:15 by Martin Rubey