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Identifier
Values
=>
Cc0002;cc-rep
[]=>0 [1]=>0 [2]=>0 [1,1]=>0 [3]=>1 [2,1]=>0 [1,1,1]=>0 [4]=>2 [3,1]=>1 [2,2]=>0 [2,1,1]=>0 [1,1,1,1]=>0 [5]=>0 [4,1]=>2 [3,2]=>1 [3,1,1]=>1 [2,2,1]=>0 [2,1,1,1]=>0 [1,1,1,1,1]=>0 [6]=>1 [5,1]=>0 [4,2]=>2 [4,1,1]=>2 [3,3]=>0 [3,2,1]=>1 [3,1,1,1]=>1 [2,2,2]=>0 [2,2,1,1]=>0 [2,1,1,1,1]=>0 [1,1,1,1,1,1]=>0 [7]=>2 [6,1]=>1 [5,2]=>0 [5,1,1]=>0 [4,3]=>3 [4,2,1]=>2 [4,1,1,1]=>2 [3,3,1]=>0 [3,2,2]=>1 [3,2,1,1]=>1 [3,1,1,1,1]=>1 [2,2,2,1]=>0 [2,2,1,1,1]=>0 [2,1,1,1,1,1]=>0 [1,1,1,1,1,1,1]=>0 [8]=>3 [7,1]=>2 [6,2]=>1 [6,1,1]=>1 [5,3]=>1 [5,2,1]=>0 [5,1,1,1]=>0 [4,4]=>0 [4,3,1]=>3 [4,2,2]=>2 [4,2,1,1]=>2 [4,1,1,1,1]=>2 [3,3,2]=>0 [3,3,1,1]=>0 [3,2,2,1]=>1 [3,2,1,1,1]=>1 [3,1,1,1,1,1]=>1 [2,2,2,2]=>0 [2,2,2,1,1]=>0 [2,2,1,1,1,1]=>0 [2,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,1,1]=>0 [9]=>1 [8,1]=>3 [7,2]=>2 [7,1,1]=>2 [6,3]=>0 [6,2,1]=>1 [6,1,1,1]=>1 [5,4]=>2 [5,3,1]=>1 [5,2,2]=>0 [5,2,1,1]=>0 [5,1,1,1,1]=>0 [4,4,1]=>0 [4,3,2]=>3 [4,3,1,1]=>3 [4,2,2,1]=>2 [4,2,1,1,1]=>2 [4,1,1,1,1,1]=>2 [3,3,3]=>1 [3,3,2,1]=>0 [3,3,1,1,1]=>0 [3,2,2,2]=>1 [3,2,2,1,1]=>1 [3,2,1,1,1,1]=>1 [3,1,1,1,1,1,1]=>1 [2,2,2,2,1]=>0 [2,2,2,1,1,1]=>0 [2,2,1,1,1,1,1]=>0 [2,1,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,1,1,1]=>0 [10]=>2 [9,1]=>1 [8,2]=>3 [8,1,1]=>3 [7,3]=>3 [7,2,1]=>2 [7,1,1,1]=>2 [6,4]=>3 [6,3,1]=>0 [6,2,2]=>1 [6,2,1,1]=>1 [6,1,1,1,1]=>1 [5,5]=>0 [5,4,1]=>2 [5,3,2]=>1 [5,3,1,1]=>1 [5,2,2,1]=>0 [5,2,1,1,1]=>0 [5,1,1,1,1,1]=>0 [4,4,2]=>0 [4,4,1,1]=>0 [4,3,3]=>2 [4,3,2,1]=>3 [4,3,1,1,1]=>3 [4,2,2,2]=>2 [4,2,2,1,1]=>2 [4,2,1,1,1,1]=>2 [4,1,1,1,1,1,1]=>2 [3,3,3,1]=>1 [3,3,2,2]=>0 [3,3,2,1,1]=>0 [3,3,1,1,1,1]=>0 [3,2,2,2,1]=>1 [3,2,2,1,1,1]=>1 [3,2,1,1,1,1,1]=>1 [3,1,1,1,1,1,1,1]=>1 [2,2,2,2,2]=>0 [2,2,2,2,1,1]=>0 [2,2,2,1,1,1,1]=>0 [2,2,1,1,1,1,1,1]=>0 [2,1,1,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,1,1,1,1]=>0 [11]=>3 [10,1]=>2 [9,2]=>1 [9,1,1]=>1 [8,3]=>2 [8,2,1]=>3 [8,1,1,1]=>3 [7,4]=>0 [7,3,1]=>3 [7,2,2]=>2 [7,2,1,1]=>2 [7,1,1,1,1]=>2 [6,5]=>1 [6,4,1]=>3 [6,3,2]=>0 [6,3,1,1]=>0 [6,2,2,1]=>1 [6,2,1,1,1]=>1 [6,1,1,1,1,1]=>1 [5,5,1]=>0 [5,4,2]=>2 [5,4,1,1]=>2 [5,3,3]=>0 [5,3,2,1]=>1 [5,3,1,1,1]=>1 [5,2,2,2]=>0 [5,2,2,1,1]=>0 [5,2,1,1,1,1]=>0 [5,1,1,1,1,1,1]=>0 [4,4,3]=>1 [4,4,2,1]=>0 [4,4,1,1,1]=>0 [4,3,3,1]=>2 [4,3,2,2]=>3 [4,3,2,1,1]=>3 [4,3,1,1,1,1]=>3 [4,2,2,2,1]=>2 [4,2,2,1,1,1]=>2 [4,2,1,1,1,1,1]=>2 [4,1,1,1,1,1,1,1]=>2 [3,3,3,2]=>1 [3,3,3,1,1]=>1 [3,3,2,2,1]=>0 [3,3,2,1,1,1]=>0 [3,3,1,1,1,1,1]=>0 [3,2,2,2,2]=>1 [3,2,2,2,1,1]=>1 [3,2,2,1,1,1,1]=>1 [3,2,1,1,1,1,1,1]=>1 [3,1,1,1,1,1,1,1,1]=>1 [2,2,2,2,2,1]=>0 [2,2,2,2,1,1,1]=>0 [2,2,2,1,1,1,1,1]=>0 [2,2,1,1,1,1,1,1,1]=>0 [2,1,1,1,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,1,1,1,1,1]=>0 [12]=>4 [11,1]=>3 [10,2]=>2 [10,1,1]=>2 [9,3]=>0 [9,2,1]=>1 [9,1,1,1]=>1 [8,4]=>1 [8,3,1]=>2 [8,2,2]=>3 [8,2,1,1]=>3 [8,1,1,1,1]=>3 [7,5]=>2 [7,4,1]=>0 [7,3,2]=>3 [7,3,1,1]=>3 [7,2,2,1]=>2 [7,2,1,1,1]=>2 [7,1,1,1,1,1]=>2 [6,6]=>0 [6,5,1]=>1 [6,4,2]=>3 [6,4,1,1]=>3 [6,3,3]=>1 [6,3,2,1]=>0 [6,3,1,1,1]=>0 [6,2,2,2]=>1 [6,2,2,1,1]=>1 [6,2,1,1,1,1]=>1 [6,1,1,1,1,1,1]=>1 [5,5,2]=>0 [5,5,1,1]=>0 [5,4,3]=>3 [5,4,2,1]=>2 [5,4,1,1,1]=>2 [5,3,3,1]=>0 [5,3,2,2]=>1 [5,3,2,1,1]=>1 [5,3,1,1,1,1]=>1 [5,2,2,2,1]=>0 [5,2,2,1,1,1]=>0 [5,2,1,1,1,1,1]=>0 [5,1,1,1,1,1,1,1]=>0 [4,4,4]=>2 [4,4,3,1]=>1 [4,4,2,2]=>0 [4,4,2,1,1]=>0 [4,4,1,1,1,1]=>0 [4,3,3,2]=>2 [4,3,3,1,1]=>2 [4,3,2,2,1]=>3 [4,3,2,1,1,1]=>3 [4,3,1,1,1,1,1]=>3 [4,2,2,2,2]=>2 [4,2,2,2,1,1]=>2 [4,2,2,1,1,1,1]=>2 [4,2,1,1,1,1,1,1]=>2 [4,1,1,1,1,1,1,1,1]=>2 [3,3,3,3]=>0 [3,3,3,2,1]=>1 [3,3,3,1,1,1]=>1 [3,3,2,2,2]=>0 [3,3,2,2,1,1]=>0 [3,3,2,1,1,1,1]=>0 [3,3,1,1,1,1,1,1]=>0 [3,2,2,2,2,1]=>1 [3,2,2,2,1,1,1]=>1 [3,2,2,1,1,1,1,1]=>1 [3,2,1,1,1,1,1,1,1]=>1 [3,1,1,1,1,1,1,1,1,1]=>1 [2,2,2,2,2,2]=>0 [2,2,2,2,2,1,1]=>0 [2,2,2,2,1,1,1,1]=>0 [2,2,2,1,1,1,1,1,1]=>0 [2,2,1,1,1,1,1,1,1,1]=>0 [2,1,1,1,1,1,1,1,1,1,1]=>0 [1,1,1,1,1,1,1,1,1,1,1,1]=>0 [5,4,3,1]=>3 [5,4,2,2]=>2 [5,4,2,1,1]=>2 [5,3,3,2]=>0 [5,3,3,1,1]=>0 [5,3,2,2,1]=>1 [4,4,3,2]=>1 [4,4,3,1,1]=>1 [4,4,2,2,1]=>0 [4,3,3,2,1]=>2 [5,4,3,2]=>3 [5,4,3,1,1]=>3 [5,4,2,2,1]=>2 [5,3,3,2,1]=>0 [4,4,3,2,1]=>1 [5,4,3,2,1]=>3
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Description
The Grundy value for the game 'Couples are forever' on an integer partition.
Two players alternately choose a part of the partition greater than two, and split it into two parts. The player facing a partition with all parts at most two looses.
References
[1] Caines, I., Gates, C., Guy, R. K., Nowakowski, R. J. Unsolved Problems: Periods in Taking and Splitting Games MathSciNet:1543453
Code
@cached_function
def statistic(la):
    """Return the Grundy value of the partition for the game 'couples are
    forever'.
    """
    def children(la):
        for i in range(len(la)):
            p = la[i]
            if p > 2:
                mu = la[:i] + la[i+1:]
                for k in range(1, (p+2)//2):
                    yield Partition(sorted(mu + [p-k, k], reverse=True))

    l = [statistic(mu) for mu in children(la)]
    i = 0
    while i in l:
        i += 1
    
    return i
Created
Apr 06, 2017 at 23:39 by Martin Rubey
Updated
Dec 31, 2017 at 14:08 by Martin Rubey