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Identifier
Values
[] => 0
[1] => 1
[2] => 2
[1,1] => 0
[3] => 3
[2,1] => 1
[1,1,1] => 1
[4] => 4
[3,1] => 2
[2,2] => 0
[2,1,1] => 0
[1,1,1,1] => 0
[5] => 5
[4,1] => 3
[3,2] => 1
[3,1,1] => 3
[2,2,1] => 1
[2,1,1,1] => 1
[1,1,1,1,1] => 1
[6] => 6
[5,1] => 4
[4,2] => 2
[4,1,1] => 2
[3,3] => 0
[3,2,1] => 0
[3,1,1,1] => 2
[2,2,2] => 2
[2,2,1,1] => 0
[2,1,1,1,1] => 0
[1,1,1,1,1,1] => 0
[7] => 7
[6,1] => 5
[5,2] => 3
[5,1,1] => 5
[4,3] => 1
[4,2,1] => 3
[4,1,1,1] => 3
[3,3,1] => 1
[3,2,2] => 3
[3,2,1,1] => 1
[3,1,1,1,1] => 3
[2,2,2,1] => 1
[2,2,1,1,1] => 1
[2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => 1
[8] => 8
[7,1] => 6
[6,2] => 4
[6,1,1] => 4
[5,3] => 2
[5,2,1] => 2
[5,1,1,1] => 4
[4,4] => 0
[4,3,1] => 0
[4,2,2] => 0
[4,2,1,1] => 2
[4,1,1,1,1] => 2
[3,3,2] => 2
[3,3,1,1] => 0
[3,2,2,1] => 0
[3,2,1,1,1] => 0
[3,1,1,1,1,1] => 2
[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] => 9
[8,1] => 7
[7,2] => 5
[7,1,1] => 7
[6,3] => 3
[6,2,1] => 5
[6,1,1,1] => 5
[5,4] => 1
[5,3,1] => 3
[5,2,2] => 1
[5,2,1,1] => 3
[5,1,1,1,1] => 5
[4,4,1] => 1
[4,3,2] => 3
[4,3,1,1] => 1
[4,2,2,1] => 3
[4,2,1,1,1] => 3
[4,1,1,1,1,1] => 3
[3,3,3] => 3
[3,3,2,1] => 1
[3,3,1,1,1] => 1
[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] => 3
[2,2,2,2,1] => 1
[2,2,2,1,1,1] => 1
[2,2,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => 1
[10] => 10
[9,1] => 8
[8,2] => 6
[8,1,1] => 6
>>> Load all 272 entries. <<<
[7,3] => 4
[7,2,1] => 4
[7,1,1,1] => 6
[6,4] => 2
[6,3,1] => 2
[6,2,2] => 6
[6,2,1,1] => 4
[6,1,1,1,1] => 4
[5,5] => 0
[5,4,1] => 0
[5,3,2] => 0
[5,3,1,1] => 2
[5,2,2,1] => 2
[5,2,1,1,1] => 2
[5,1,1,1,1,1] => 4
[4,4,2] => 2
[4,4,1,1] => 0
[4,3,3] => 2
[4,3,2,1] => 0
[4,3,1,1,1] => 0
[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] => 2
[3,3,2,2] => 0
[3,3,2,1,1] => 0
[3,3,1,1,1,1] => 0
[3,2,2,2,1] => 0
[3,2,2,1,1,1] => 0
[3,2,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1] => 2
[2,2,2,2,2] => 2
[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] => 11
[10,1] => 9
[9,2] => 7
[9,1,1] => 9
[8,3] => 5
[8,2,1] => 7
[8,1,1,1] => 7
[7,4] => 3
[7,3,1] => 5
[7,2,2] => 7
[7,2,1,1] => 5
[7,1,1,1,1] => 7
[6,5] => 1
[6,4,1] => 3
[6,3,2] => 1
[6,3,1,1] => 3
[6,2,2,1] => 5
[6,2,1,1,1] => 5
[6,1,1,1,1,1] => 5
[5,5,1] => 1
[5,4,2] => 3
[5,4,1,1] => 1
[5,3,3] => 1
[5,3,2,1] => 3
[5,3,1,1,1] => 3
[5,2,2,2] => 3
[5,2,2,1,1] => 3
[5,2,1,1,1,1] => 3
[5,1,1,1,1,1,1] => 5
[4,4,3] => 3
[4,4,2,1] => 1
[4,4,1,1,1] => 1
[4,3,3,1] => 3
[4,3,2,2] => 1
[4,3,2,1,1] => 1
[4,3,1,1,1,1] => 1
[4,2,2,2,1] => 3
[4,2,2,1,1,1] => 3
[4,2,1,1,1,1,1] => 3
[4,1,1,1,1,1,1,1] => 3
[3,3,3,2] => 1
[3,3,3,1,1] => 3
[3,3,2,2,1] => 1
[3,3,2,1,1,1] => 1
[3,3,1,1,1,1,1] => 1
[3,2,2,2,2] => 3
[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] => 3
[2,2,2,2,2,1] => 1
[2,2,2,2,1,1,1] => 1
[2,2,2,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 12
[11,1] => 10
[10,2] => 8
[10,1,1] => 8
[9,3] => 6
[9,2,1] => 6
[9,1,1,1] => 8
[8,4] => 4
[8,3,1] => 4
[8,2,2] => 4
[8,2,1,1] => 6
[8,1,1,1,1] => 6
[7,5] => 2
[7,4,1] => 2
[7,3,2] => 6
[7,3,1,1] => 4
[7,2,2,1] => 4
[7,2,1,1,1] => 4
[7,1,1,1,1,1] => 6
[6,6] => 0
[6,5,1] => 0
[6,4,2] => 0
[6,4,1,1] => 2
[6,3,3] => 0
[6,3,2,1] => 2
[6,3,1,1,1] => 2
[6,2,2,2] => 4
[6,2,2,1,1] => 4
[6,2,1,1,1,1] => 4
[6,1,1,1,1,1,1] => 4
[5,5,2] => 2
[5,5,1,1] => 0
[5,4,3] => 2
[5,4,2,1] => 0
[5,4,1,1,1] => 0
[5,3,3,1] => 0
[5,3,2,2] => 2
[5,3,2,1,1] => 2
[5,3,1,1,1,1] => 2
[5,2,2,2,1] => 2
[5,2,2,1,1,1] => 2
[5,2,1,1,1,1,1] => 2
[5,1,1,1,1,1,1,1] => 4
[4,4,4] => 4
[4,4,3,1] => 2
[4,4,2,2] => 0
[4,4,2,1,1] => 0
[4,4,1,1,1,1] => 0
[4,3,3,2] => 0
[4,3,3,1,1] => 2
[4,3,2,2,1] => 0
[4,3,2,1,1,1] => 0
[4,3,1,1,1,1,1] => 0
[4,2,2,2,2] => 0
[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] => 0
[3,3,3,1,1,1] => 2
[3,3,2,2,2] => 2
[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] => 0
[3,2,2,2,1,1,1] => 0
[3,2,2,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1,1] => 2
[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
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Description
The Grundy value for the game of removing cells of a row in an integer partition.
Two players alternately remove any positive number of cells in a row of the Ferrers diagram of an integer partition, such that the result is still a Ferrers diagram. The player facing the empty partition looses.
References
[1] Row, A. Ordered Nim game MathOverflow:286925
Code
@cached_function
def statistic(la):
    """Return the Grundy value of the partition for the game
    where one may remove a positive number of cells in one row.
    """
    def children(la):
        if len(la) == 0:
            return
        for i in range(len(la)-1):
            for j in range(1, la[i]-la[i+1]+1):
                mu = [la[k] if k != i else la[k]-j for k in range(len(la))]
                yield Partition(mu)
        for j in range(1, la[-1]+1):
            mu = la[:-1] + [la[-1]-j]
            yield Partition(mu)
    
    l = [statistic(mu) for mu in children(la)]
    i = 0
    while i in l:
        i += 1
    
    return i
Created
Nov 25, 2017 at 13:45 by Martin Rubey
Updated
Nov 25, 2017 at 13:45 by Martin Rubey