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Identifier

St001384: Integer partitions ⟶ ℤ

Values

[1] => 0

[2] => 1

[1,1] => 1

[3] => 2

[2,1] => 0

[1,1,1] => 2

[4] => 3

[3,1] => 1

[2,2] => 1

[2,1,1] => 1

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

[5] => 4

[4,1] => 2

[3,2] => 2

[3,1,1] => 2

[2,2,1] => 2

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

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

[6] => 5

[5,1] => 3

[4,2] => 3

[4,1,1] => 3

[3,3] => 3

[3,2,1] => 0

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

[2,2,2] => 3

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

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

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

[7] => 6

[6,1] => 4

[5,2] => 4

[5,1,1] => 4

[4,3] => 4

[4,2,1] => 1

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

[3,3,1] => 1

[3,2,2] => 1

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

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

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

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

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

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

[8] => 7

[7,1] => 5

[6,2] => 5

[6,1,1] => 5

[5,3] => 5

[5,2,1] => 2

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

[4,4] => 5

[4,3,1] => 2

[4,2,2] => 2

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

[4,1,1,1,1] => 5

[3,3,2] => 2

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,1,1] => 7

[9] => 8

[8,1] => 6

[7,2] => 6

[7,1,1] => 6

[6,3] => 6

[6,2,1] => 3

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

[5,4] => 6

[5,3,1] => 3

[5,2,2] => 3

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

[5,1,1,1,1] => 6

[4,4,1] => 3

[4,3,2] => 3

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

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

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

[4,1,1,1,1,1] => 6

[3,3,3] => 3

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

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

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

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

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

[3,1,1,1,1,1,1] => 6

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

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

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

[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] => 7

[8,1,1] => 7

[7,3] => 7

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Description

The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.

References

[1] The side length of the largest staircase partition fitting into a partition. St000783

Code

def statistic(L):
    St000783 = min(p + i for i, p in enumerate(L + [0]))
    return sum(L) - St000783*(St000783+Integer(1)) // 2

Created

Apr 19, 2019 at 20:43 by Christian Stump

Updated

Apr 19, 2019 at 20:43 by Christian Stump