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Identifier

St000759: Integer partitions ⟶ ℤ

Values

[] => 1

[1] => 2

[2] => 1

[1,1] => 2

[3] => 1

[2,1] => 3

[1,1,1] => 2

[4] => 1

[3,1] => 2

[2,2] => 1

[2,1,1] => 3

[1,1,1,1] => 2

[5] => 1

[4,1] => 2

[3,2] => 1

[3,1,1] => 2

[2,2,1] => 3

[2,1,1,1] => 3

[1,1,1,1,1] => 2

[6] => 1

[5,1] => 2

[4,2] => 1

[4,1,1] => 2

[3,3] => 1

[3,2,1] => 4

[3,1,1,1] => 2

[2,2,2] => 1

[2,2,1,1] => 3

[2,1,1,1,1] => 3

[1,1,1,1,1,1] => 2

[7] => 1

[6,1] => 2

[5,2] => 1

[5,1,1] => 2

[4,3] => 1

[4,2,1] => 3

[4,1,1,1] => 2

[3,3,1] => 2

[3,2,2] => 1

[3,2,1,1] => 4

[3,1,1,1,1] => 2

[2,2,2,1] => 3

[2,2,1,1,1] => 3

[2,1,1,1,1,1] => 3

[1,1,1,1,1,1,1] => 2

[8] => 1

[7,1] => 2

[6,2] => 1

[6,1,1] => 2

[5,3] => 1

[5,2,1] => 3

[5,1,1,1] => 2

[4,4] => 1

[4,3,1] => 2

[4,2,2] => 1

[4,2,1,1] => 3

[4,1,1,1,1] => 2

[3,3,2] => 1

[3,3,1,1] => 2

[3,2,2,1] => 4

[3,2,1,1,1] => 4

[3,1,1,1,1,1] => 2

[2,2,2,2] => 1

[2,2,2,1,1] => 3

[2,2,1,1,1,1] => 3

[2,1,1,1,1,1,1] => 3

[1,1,1,1,1,1,1,1] => 2

[9] => 1

[8,1] => 2

[7,2] => 1

[7,1,1] => 2

[6,3] => 1

[6,2,1] => 3

[6,1,1,1] => 2

[5,4] => 1

[5,3,1] => 2

[5,2,2] => 1

[5,2,1,1] => 3

[5,1,1,1,1] => 2

[4,4,1] => 2

[4,3,2] => 1

[4,3,1,1] => 2

[4,2,2,1] => 3

[4,2,1,1,1] => 3

[4,1,1,1,1,1] => 2

[3,3,3] => 1

[3,3,2,1] => 4

[3,3,1,1,1] => 2

[3,2,2,2] => 1

[3,2,2,1,1] => 4

[3,2,1,1,1,1] => 4

[3,1,1,1,1,1,1] => 2

[2,2,2,2,1] => 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

[1,1,1,1,1,1,1,1,1] => 2

[10] => 1

[9,1] => 2

[8,2] => 1

[8,1,1] => 2

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Description

The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].

References

[1] Hitczenko, Paweł, Louchard, G. Distinctness of compositions of an integer: a probabilistic analysis MathSciNet:1871561
[2] Triangle read by rows: T(n,k) is the number of partitions of n having least gap k. OEIS:A264401
[3] Hopkins, B., Sellers, J. A., Yee, A. J. Combinatorial Perspectives on the Crank and Mex Partition Statistics arXiv:2108.09414

Code

def statistic(pi):
    return min(set(range(1,len(pi)+2)).difference(set(pi)))

Created

Apr 08, 2017 at 22:40 by Martin Rubey

Updated

Aug 24, 2021 at 12:59 by Martin Rubey