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Identifier
Values
=>
Cc0002;cc-rep
[]=>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 [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