- FindStat

Identifier

St000480: Integer partitions ⟶ ℤ

Values

[] => 0

[1] => 0

[2] => 1

[1,1] => 0

[3] => 1

[2,1] => 1

[1,1,1] => 0

[4] => 1

[3,1] => 1

[2,2] => 1

[2,1,1] => 1

[1,1,1,1] => 0

[5] => 1

[4,1] => 1

[3,2] => 1

[3,1,1] => 1

[2,2,1] => 1

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

[1,1,1,1,1] => 0

[6] => 1

[5,1] => 1

[4,2] => 2

[4,1,1] => 1

[3,3] => 1

[3,2,1] => 2

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

[2,2,2] => 1

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

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

[1,1,1,1,1,1] => 0

[7] => 1

[6,1] => 1

[5,2] => 2

[5,1,1] => 1

[4,3] => 1

[4,2,1] => 2

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

[3,3,1] => 1

[3,2,2] => 1

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

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

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

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

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

[1,1,1,1,1,1,1] => 0

[8] => 1

[7,1] => 1

[6,2] => 2

[6,1,1] => 1

[5,3] => 2

[5,2,1] => 2

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

[4,4] => 1

[4,3,1] => 1

[4,2,2] => 2

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

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

[3,3,2] => 1

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,1,1] => 0

[9] => 1

[8,1] => 1

[7,2] => 2

[7,1,1] => 1

[6,3] => 2

[6,2,1] => 2

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

[5,4] => 1

[5,3,1] => 2

[5,2,2] => 2

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

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

[4,4,1] => 1

[4,3,2] => 2

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

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

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

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

[3,3,3] => 1

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,1,1,1] => 0

[10] => 1

[9,1] => 1

[8,2] => 2

[8,1,1] => 1

>>> Load all 272 entries. <<<

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Description

The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$ \frac{1}{2}(\sqrt{1+8n}-3) $$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.

References

[1] Brylawski, T. The lattice of integer partitions MathSciNet:0325405

Code

@cached_function
def P(k):
    return posets.IntegerPartitionsDominanceOrder(k)

def statistic(pi):
    Q = P(pi.size())
    return len(Q.lower_covers(Q(pi)))

Created

May 09, 2016 at 09:22 by Martin Rubey

Updated

Oct 29, 2017 at 21:36 by Martin Rubey