- FindStat

Identifier

St000384: Integer partitions ⟶ ℤ

Values

[] => 0

[1] => 1

[2] => 2

[1,1] => 2

[3] => 3

[2,1] => 2

[1,1,1] => 3

[4] => 4

[3,1] => 3

[2,2] => 3

[2,1,1] => 3

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

[5] => 5

[4,1] => 4

[3,2] => 3

[3,1,1] => 3

[2,2,1] => 3

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

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

[6] => 6

[5,1] => 5

[4,2] => 4

[4,1,1] => 4

[3,3] => 4

[3,2,1] => 3

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

[2,2,2] => 4

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

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

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

[7] => 7

[6,1] => 6

[5,2] => 5

[5,1,1] => 5

[4,3] => 4

[4,2,1] => 4

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

[3,3,1] => 4

[3,2,2] => 4

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

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

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

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

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

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

[8] => 8

[7,1] => 7

[6,2] => 6

[6,1,1] => 6

[5,3] => 5

[5,2,1] => 5

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

[4,4] => 5

[4,3,1] => 4

[4,2,2] => 4

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

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

[3,3,2] => 4

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

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

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

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

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

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

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

[2,1,1,1,1,1,1] => 7

[1,1,1,1,1,1,1,1] => 8

[9] => 9

[8,1] => 8

[7,2] => 7

[7,1,1] => 7

[6,3] => 6

[6,2,1] => 6

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

[5,4] => 5

[5,3,1] => 5

[5,2,2] => 5

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

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

[4,4,1] => 5

[4,3,2] => 4

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

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

[4,2,1,1,1] => 5

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

[3,3,3] => 5

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

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

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

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

[3,2,1,1,1,1] => 6

[3,1,1,1,1,1,1] => 7

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

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

[2,2,1,1,1,1,1] => 7

[2,1,1,1,1,1,1,1] => 8

[1,1,1,1,1,1,1,1,1] => 9

[10] => 10

[9,1] => 9

[8,2] => 8

[8,1,1] => 8

>>> Load all 288 entries. <<<

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Description

The maximal part of the shifted composition of an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding $i-1$ to the $i$-th part.
The statistic is then $\operatorname{max}_i\{ \lambda_i + i - 1 \}$.
See also St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition..

Code

def statistic(p):
    if len(p) == 0:
        return 0
    return max( p[i]+i for i in range(len(p)) )

Created

Feb 09, 2016 at 12:23 by Christian Stump

Updated

May 24, 2018 at 13:24 by Martin Rubey