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Identifier

St001915: Binary words ⟶ ℤ

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

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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.

References

[1] Brandt, Jør. Cycles of partitions 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)

@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)))

Created

Aug 16, 2023 at 08:53 by Martin Rubey

Updated

Aug 16, 2023 at 08:53 by Martin Rubey