- FindStat

Identifier

St000752: Integer partitions ⟶ ℤ

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