- FindStat

Identifier

St001055: Integer partitions ⟶ ℤ

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