- FindStat

Identifier

St000145: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [1]=>0
[2]=>1
[1,1]=>-1
[3]=>2
[2,1]=>0
[1,1,1]=>-2
[4]=>3
[3,1]=>1
[2,2]=>0
[2,1,1]=>-1
[1,1,1,1]=>-3
[5]=>4
[4,1]=>2
[3,2]=>1
[3,1,1]=>0
[2,2,1]=>-1
[2,1,1,1]=>-2
[1,1,1,1,1]=>-4
[6]=>5
[5,1]=>3
[4,2]=>2
[4,1,1]=>1
[3,3]=>1
[3,2,1]=>0
[3,1,1,1]=>-1
[2,2,2]=>-1
[2,2,1,1]=>-2
[2,1,1,1,1]=>-3
[1,1,1,1,1,1]=>-5
[7]=>6
[6,1]=>4
[5,2]=>3
[5,1,1]=>2
[4,3]=>2
[4,2,1]=>1
[4,1,1,1]=>0
[3,3,1]=>0
[3,2,2]=>0
[3,2,1,1]=>-1
[3,1,1,1,1]=>-2
[2,2,2,1]=>-2
[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]=>5
[6,2]=>4
[6,1,1]=>3
[5,3]=>3
[5,2,1]=>2
[5,1,1,1]=>1
[4,4]=>2
[4,3,1]=>1
[4,2,2]=>1
[4,2,1,1]=>0
[4,1,1,1,1]=>-1
[3,3,2]=>0
[3,3,1,1]=>-1
[3,2,2,1]=>-1
[3,2,1,1,1]=>-2
[3,1,1,1,1,1]=>-3
[2,2,2,2]=>-2
[2,2,2,1,1]=>-3
[2,2,1,1,1,1]=>-4
[2,1,1,1,1,1,1]=>-5
[1,1,1,1,1,1,1,1]=>-7
[9]=>8
[8,1]=>6
[7,2]=>5
[7,1,1]=>4
[6,3]=>4
[6,2,1]=>3
[6,1,1,1]=>2
[5,4]=>3
[5,3,1]=>2
[5,2,2]=>2
[5,2,1,1]=>1
[5,1,1,1,1]=>0
[4,4,1]=>1
[4,3,2]=>1
[4,3,1,1]=>0
[4,2,2,1]=>0
[4,2,1,1,1]=>-1
[4,1,1,1,1,1]=>-2
[3,3,3]=>0
[3,3,2,1]=>-1
[3,3,1,1,1]=>-2
[3,2,2,2]=>-1
[3,2,2,1,1]=>-2
[3,2,1,1,1,1]=>-3
[3,1,1,1,1,1,1]=>-4
[2,2,2,2,1]=>-3
[2,2,2,1,1,1]=>-4
[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]=>7
[8,2]=>6
[8,1,1]=>5
[7,3]=>5
[7,2,1]=>4
[7,1,1,1]=>3
[6,4]=>4
[6,3,1]=>3
[6,2,2]=>3
[6,2,1,1]=>2
[6,1,1,1,1]=>1
[5,5]=>3
[5,4,1]=>2
[5,3,2]=>2
[5,3,1,1]=>1
[5,2,2,1]=>1
[5,2,1,1,1]=>0
[5,1,1,1,1,1]=>-1
[4,4,2]=>1
[4,4,1,1]=>0
[4,3,3]=>1
[4,3,2,1]=>0
[4,3,1,1,1]=>-1
[4,2,2,2]=>0
[4,2,2,1,1]=>-1
[4,2,1,1,1,1]=>-2
[4,1,1,1,1,1,1]=>-3
[3,3,3,1]=>-1
[3,3,2,2]=>-1
[3,3,2,1,1]=>-2
[3,3,1,1,1,1]=>-3
[3,2,2,2,1]=>-2
[3,2,2,1,1,1]=>-3
[3,2,1,1,1,1,1]=>-4
[3,1,1,1,1,1,1,1]=>-5
[2,2,2,2,2]=>-3
[2,2,2,2,1,1]=>-4
[2,2,2,1,1,1,1]=>-5
[2,2,1,1,1,1,1,1]=>-6
[2,1,1,1,1,1,1,1,1]=>-7
[1,1,1,1,1,1,1,1,1,1]=>-9
                    

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Description

The Dyson rank of a partition.
This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences $$p(5n+4) \equiv 0 \pmod 5$$ and $$p(7n+6) \equiv 0 \pmod 7.$$

References

[1] Dyson, F. J. Some guesses in the theory of partitions MathSciNet:3077150

Code

def statistic(L):
    return L[0] - len(L)

Created

Jul 03, 2013 at 14:34 by Olivier Mallet

Updated

May 29, 2015 at 16:57 by Martin Rubey