- FindStat

Identifier

St000681: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [2]=>1
[1,1]=>1
[3]=>2
[2,1]=>0
[1,1,1]=>2
[4]=>3
[3,1]=>3
[2,2]=>2
[2,1,1]=>3
[1,1,1,1]=>3
[5]=>4
[4,1]=>2
[3,2]=>0
[3,1,1]=>0
[2,2,1]=>0
[2,1,1,1]=>2
[1,1,1,1,1]=>4
[6]=>5
[5,1]=>5
[4,2]=>4
[4,1,1]=>1
[3,3]=>4
[3,2,1]=>1
[3,1,1,1]=>1
[2,2,2]=>4
[2,2,1,1]=>4
[2,1,1,1,1]=>5
[1,1,1,1,1,1]=>5
[7]=>6
[6,1]=>4
[5,2]=>3
[5,1,1]=>6
[4,3]=>0
[4,2,1]=>5
[4,1,1,1]=>0
[3,3,1]=>3
[3,2,2]=>3
[3,2,1,1]=>5
[3,1,1,1,1]=>6
[2,2,2,1]=>0
[2,2,1,1,1]=>3
[2,1,1,1,1,1]=>4
[1,1,1,1,1,1,1]=>6
[8]=>7
[7,1]=>7
[6,2]=>6
[6,1,1]=>7
[5,3]=>6
[5,2,1]=>7
[5,1,1,1]=>7
[4,4]=>5
[4,3,1]=>4
[4,2,2]=>0
[4,2,1,1]=>1
[4,1,1,1,1]=>7
[3,3,2]=>1
[3,3,1,1]=>0
[3,2,2,1]=>4
[3,2,1,1,1]=>7
[3,1,1,1,1,1]=>7
[2,2,2,2]=>5
[2,2,2,1,1]=>6
[2,2,1,1,1,1]=>6
[2,1,1,1,1,1,1]=>7
[1,1,1,1,1,1,1,1]=>7
[9]=>8
[8,1]=>6
[7,2]=>5
[7,1,1]=>4
[6,3]=>3
[6,2,1]=>3
[6,1,1,1]=>6
[5,4]=>0
[5,3,1]=>1
[5,2,2]=>1
[5,2,1,1]=>0
[5,1,1,1,1]=>0
[4,4,1]=>6
[4,3,2]=>5
[4,3,1,1]=>2
[4,2,2,1]=>2
[4,2,1,1,1]=>0
[4,1,1,1,1,1]=>6
[3,3,3]=>5
[3,3,2,1]=>5
[3,3,1,1,1]=>1
[3,2,2,2]=>6
[3,2,2,1,1]=>1
[3,2,1,1,1,1]=>3
[3,1,1,1,1,1,1]=>4
[2,2,2,2,1]=>0
[2,2,2,1,1,1]=>3
[2,2,1,1,1,1,1]=>5
[2,1,1,1,1,1,1,1]=>6
[1,1,1,1,1,1,1,1,1]=>8
[10]=>9
[9,1]=>9
[8,2]=>8
[8,1,1]=>5
[7,3]=>8
[7,2,1]=>8
[7,1,1,1]=>5
[6,4]=>7
[6,3,1]=>6
[6,2,2]=>8
[6,2,1,1]=>2
[6,1,1,1,1]=>1
[5,5]=>7
[5,4,1]=>5
[5,3,2]=>0
[5,3,1,1]=>5
[5,2,2,1]=>5
[5,2,1,1,1]=>1
[5,1,1,1,1,1]=>1
[4,4,2]=>7
[4,4,1,1]=>7
[4,3,3]=>6
[4,3,2,1]=>1
[4,3,1,1,1]=>5
[4,2,2,2]=>7
[4,2,2,1,1]=>5
[4,2,1,1,1,1]=>2
[4,1,1,1,1,1,1]=>5
[3,3,3,1]=>6
[3,3,2,2]=>7
[3,3,2,1,1]=>0
[3,3,1,1,1,1]=>8
[3,2,2,2,1]=>5
[3,2,2,1,1,1]=>6
[3,2,1,1,1,1,1]=>8
[3,1,1,1,1,1,1,1]=>5
[2,2,2,2,2]=>7
[2,2,2,2,1,1]=>7
[2,2,2,1,1,1,1]=>8
[2,2,1,1,1,1,1,1]=>8
[2,1,1,1,1,1,1,1,1]=>9
[1,1,1,1,1,1,1,1,1,1]=>9
[11]=>10
[10,1]=>8
[9,2]=>7
[9,1,1]=>10
[8,3]=>5
[8,2,1]=>4
[8,1,1,1]=>4
[7,4]=>3
[7,3,1]=>5
[7,2,2]=>7
[7,2,1,1]=>7
[7,1,1,1,1]=>2
[6,5]=>0
[6,4,1]=>1
[6,3,2]=>9
[6,3,1,1]=>1
[6,2,2,1]=>1
[6,2,1,1,1]=>8
[6,1,1,1,1,1]=>0
[5,5,1]=>8
[5,4,2]=>3
[5,4,1,1]=>1
[5,3,3]=>3
[5,3,2,1]=>2
[5,3,1,1,1]=>2
[5,2,2,2]=>2
[5,2,2,1,1]=>2
[5,2,1,1,1,1]=>8
[5,1,1,1,1,1,1]=>2
[4,4,3]=>8
[4,4,2,1]=>4
[4,4,1,1,1]=>2
[4,3,3,1]=>4
[4,3,2,2]=>4
[4,3,2,1,1]=>2
[4,3,1,1,1,1]=>1
[4,2,2,2,1]=>1
[4,2,2,1,1,1]=>1
[4,2,1,1,1,1,1]=>7
[4,1,1,1,1,1,1,1]=>4
[3,3,3,2]=>8
[3,3,3,1,1]=>3
[3,3,2,2,1]=>3
[3,3,2,1,1,1]=>9
[3,3,1,1,1,1,1]=>7
[3,2,2,2,2]=>8
[3,2,2,2,1,1]=>1
[3,2,2,1,1,1,1]=>5
[3,2,1,1,1,1,1,1]=>4
[3,1,1,1,1,1,1,1,1]=>10
[2,2,2,2,2,1]=>0
[2,2,2,2,1,1,1]=>3
[2,2,2,1,1,1,1,1]=>5
[2,2,1,1,1,1,1,1,1]=>7
[2,1,1,1,1,1,1,1,1,1]=>8
[1,1,1,1,1,1,1,1,1,1,1]=>10
[12]=>11
[11,1]=>11
[10,2]=>10
[10,1,1]=>11
[9,3]=>10
[9,2,1]=>6
[9,1,1,1]=>11
[8,4]=>9
[8,3,1]=>8
[8,2,2]=>6
[8,2,1,1]=>6
[8,1,1,1,1]=>3
[7,5]=>9
[7,4,1]=>7
[7,3,2]=>3
[7,3,1,1]=>9
[7,2,2,1]=>8
[7,2,1,1,1]=>9
[7,1,1,1,1,1]=>3
[6,6]=>8
[6,5,1]=>4
[6,4,2]=>0
[6,4,1,1]=>8
[6,3,3]=>0
[6,3,2,1]=>4
[6,3,1,1,1]=>0
[6,2,2,2]=>0
[6,2,2,1,1]=>0
[6,2,1,1,1,1]=>1
[6,1,1,1,1,1,1]=>3
[5,5,2]=>5
[5,5,1,1]=>2
[5,4,3]=>7
[5,4,2,1]=>6
[5,4,1,1,1]=>6
[5,3,3,1]=>1
[5,3,2,2]=>1
[5,3,2,1,1]=>1
[5,3,1,1,1,1]=>0
[5,2,2,2,1]=>6
[5,2,2,1,1,1]=>0
[5,2,1,1,1,1,1]=>9
[5,1,1,1,1,1,1,1]=>3
[4,4,4]=>9
[4,4,3,1]=>1
[4,4,2,2]=>1
[4,4,2,1,1]=>1
[4,4,1,1,1,1]=>0
[4,3,3,2]=>1
[4,3,3,1,1]=>1
[4,3,2,2,1]=>6
[4,3,2,1,1,1]=>4
[4,3,1,1,1,1,1]=>8
[4,2,2,2,2]=>2
[4,2,2,2,1,1]=>8
[4,2,2,1,1,1,1]=>9
[4,2,1,1,1,1,1,1]=>6
[4,1,1,1,1,1,1,1,1]=>11
[3,3,3,3]=>9
[3,3,3,2,1]=>7
[3,3,3,1,1,1]=>0
[3,3,2,2,2]=>5
[3,3,2,2,1,1]=>0
[3,3,2,1,1,1,1]=>3
[3,3,1,1,1,1,1,1]=>6
[3,2,2,2,2,1]=>4
[3,2,2,2,1,1,1]=>7
[3,2,2,1,1,1,1,1]=>8
[3,2,1,1,1,1,1,1,1]=>6
[3,1,1,1,1,1,1,1,1,1]=>11
[2,2,2,2,2,2]=>8
[2,2,2,2,2,1,1]=>9
[2,2,2,2,1,1,1,1]=>9
[2,2,2,1,1,1,1,1,1]=>10
[2,2,1,1,1,1,1,1,1,1]=>10
[2,1,1,1,1,1,1,1,1,1,1]=>11
[1,1,1,1,1,1,1,1,1,1,1,1]=>11
                    

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Description

The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.

References

[1] wikipedia:Chomp

Code

@cached_function
def statistic(pi):
    """Return the Grundy value of the Ferrers diagram pi for Chomp.

    sage: statistic(Partition([1]))
    0

    sage: statistic(Partition([1,1]))
    1

    sage: statistic(Partition([2,1]))
    0
    """
    def children(la):
        for row, col in la.cells():
            if (row,col)  != (0,0):
                yield Partition(la[:row] + [min(r, col) for r in la[row:]])

    pi = Partition(pi)
    if pi.size() == 0:
        return
    elif pi.size() == 1:
        return 0
    else:
        l = [statistic(la) for la in children(pi)]
        i = 0
        while i in l:
            i += 1
        return i

Created

Jan 05, 2017 at 20:33 by Martin Rubey

Updated

Jan 05, 2017 at 20:33 by Martin Rubey