- FindStat

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Identifier

St001714: Integer partitions ⟶ ℤ

Values

[1] => 0

[2] => 0

[1,1] => 1

[3] => 0

[2,1] => 0

[1,1,1] => 2

[4] => 0

[3,1] => 0

[2,2] => 0

[2,1,1] => 2

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

[5] => 0

[4,1] => 0

[3,2] => 0

[3,1,1] => 1

[2,2,1] => 1

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

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

[6] => 0

[5,1] => 0

[4,2] => 0

[4,1,1] => 1

[3,3] => 0

[3,2,1] => 0

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

[2,2,2] => 1

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

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

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

[7] => 0

[6,1] => 0

[5,2] => 0

[5,1,1] => 1

[4,3] => 0

[4,2,1] => 0

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

[3,3,1] => 0

[3,2,2] => 1

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

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

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

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

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

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

[8] => 0

[7,1] => 0

[6,2] => 0

[6,1,1] => 1

[5,3] => 0

[5,2,1] => 0

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

[4,4] => 0

[4,3,1] => 0

[4,2,2] => 0

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

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

[3,3,2] => 0

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

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

[3,2,1,1,1] => 7

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

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

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

[2,2,1,1,1,1] => 10

[2,1,1,1,1,1,1] => 10

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

[9] => 0

[8,1] => 0

[7,2] => 0

[7,1,1] => 1

[6,3] => 0

[6,2,1] => 0

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

[5,4] => 0

[5,3,1] => 0

[5,2,2] => 0

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

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

[4,4,1] => 0

[4,3,2] => 0

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

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

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

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

[3,3,3] => 0

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

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

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

[3,2,2,1,1] => 8

[3,2,1,1,1,1] => 11

[3,1,1,1,1,1,1] => 9

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

[2,2,2,1,1,1] => 11

[2,2,1,1,1,1,1] => 13

[2,1,1,1,1,1,1,1] => 12

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

[10] => 0

[9,1] => 0

[8,2] => 0

[8,1,1] => 1

[7,3] => 0

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Description

The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.

References

[1] Chow, T. Y., Fan, C. K., Goemans, M. X., Vondrak, J. Wide partitions, Latin tableaux, and Rota's basis conjecture MathSciNet:2001618

Code

def statistic(la):
    c = 0
    for mu in Subsets(la, submultiset=True):
        mu = Partition(mu)
        if not mu.dominates(mu.conjugate()):
            c += 1
    return c

Created

Apr 23, 2021 at 09:54 by Martin Rubey

Updated

Apr 23, 2021 at 09:54 by Martin Rubey