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Identifier
Values
[] => 0
[1] => 0
[2] => 0
[1,1] => 1
[3] => 0
[2,1] => 1
[1,1,1] => 1
[4] => 0
[3,1] => 1
[2,2] => 1
[2,1,1] => 1
[1,1,1,1] => 1
[5] => 0
[4,1] => 1
[3,2] => 1
[3,1,1] => 1
[2,2,1] => 1
[2,1,1,1] => 1
[1,1,1,1,1] => 1
[6] => 0
[5,1] => 1
[4,2] => 1
[4,1,1] => 1
[3,3] => 1
[3,2,1] => 2
[3,1,1,1] => 1
[2,2,2] => 1
[2,2,1,1] => 2
[2,1,1,1,1] => 1
[1,1,1,1,1,1] => 1
[7] => 0
[6,1] => 1
[5,2] => 1
[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] => 2
[2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => 1
[8] => 0
[7,1] => 1
[6,2] => 1
[6,1,1] => 1
[5,3] => 1
[5,2,1] => 2
[5,1,1,1] => 1
[4,4] => 1
[4,3,1] => 2
[4,2,2] => 1
[4,2,1,1] => 2
[4,1,1,1,1] => 1
[3,3,2] => 1
[3,3,1,1] => 2
[3,2,2,1] => 1
[3,2,1,1,1] => 2
[3,1,1,1,1,1] => 1
[2,2,2,2] => 1
[2,2,2,1,1] => 2
[2,2,1,1,1,1] => 2
[2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => 1
[9] => 0
[8,1] => 1
[7,2] => 1
[7,1,1] => 1
[6,3] => 1
[6,2,1] => 2
[6,1,1,1] => 1
[5,4] => 1
[5,3,1] => 2
[5,2,2] => 1
[5,2,1,1] => 2
[5,1,1,1,1] => 1
[4,4,1] => 1
[4,3,2] => 2
[4,3,1,1] => 2
[4,2,2,1] => 1
[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] => 2
[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] => 2
[2,2,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => 1
[10] => 0
[9,1] => 1
[8,2] => 1
[8,1,1] => 1
>>> Load all 272 entries. <<<
[7,3] => 1
[7,2,1] => 2
[7,1,1,1] => 1
[6,4] => 1
[6,3,1] => 2
[6,2,2] => 1
[6,2,1,1] => 2
[6,1,1,1,1] => 1
[5,5] => 1
[5,4,1] => 2
[5,3,2] => 2
[5,3,1,1] => 2
[5,2,2,1] => 1
[5,2,1,1,1] => 2
[5,1,1,1,1,1] => 1
[4,4,2] => 1
[4,4,1,1] => 2
[4,3,3] => 1
[4,3,2,1] => 3
[4,3,1,1,1] => 2
[4,2,2,2] => 1
[4,2,2,1,1] => 2
[4,2,1,1,1,1] => 2
[4,1,1,1,1,1,1] => 1
[3,3,3,1] => 1
[3,3,2,2] => 2
[3,3,2,1,1] => 2
[3,3,1,1,1,1] => 2
[3,2,2,2,1] => 1
[3,2,2,1,1,1] => 2
[3,2,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1] => 1
[2,2,2,2,2] => 1
[2,2,2,2,1,1] => 2
[2,2,2,1,1,1,1] => 2
[2,2,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 0
[10,1] => 1
[9,2] => 1
[9,1,1] => 1
[8,3] => 1
[8,2,1] => 2
[8,1,1,1] => 1
[7,4] => 1
[7,3,1] => 2
[7,2,2] => 1
[7,2,1,1] => 2
[7,1,1,1,1] => 1
[6,5] => 1
[6,4,1] => 2
[6,3,2] => 2
[6,3,1,1] => 2
[6,2,2,1] => 1
[6,2,1,1,1] => 2
[6,1,1,1,1,1] => 1
[5,5,1] => 1
[5,4,2] => 2
[5,4,1,1] => 2
[5,3,3] => 1
[5,3,2,1] => 3
[5,3,1,1,1] => 2
[5,2,2,2] => 1
[5,2,2,1,1] => 2
[5,2,1,1,1,1] => 2
[5,1,1,1,1,1,1] => 1
[4,4,3] => 1
[4,4,2,1] => 2
[4,4,1,1,1] => 2
[4,3,3,1] => 1
[4,3,2,2] => 2
[4,3,2,1,1] => 3
[4,3,1,1,1,1] => 2
[4,2,2,2,1] => 1
[4,2,2,1,1,1] => 2
[4,2,1,1,1,1,1] => 2
[4,1,1,1,1,1,1,1] => 1
[3,3,3,2] => 1
[3,3,3,1,1] => 2
[3,3,2,2,1] => 2
[3,3,2,1,1,1] => 2
[3,3,1,1,1,1,1] => 2
[3,2,2,2,2] => 1
[3,2,2,2,1,1] => 2
[3,2,2,1,1,1,1] => 2
[3,2,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,1] => 1
[2,2,2,2,1,1,1] => 2
[2,2,2,1,1,1,1,1] => 2
[2,2,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 0
[11,1] => 1
[10,2] => 1
[10,1,1] => 1
[9,3] => 1
[9,2,1] => 2
[9,1,1,1] => 1
[8,4] => 1
[8,3,1] => 2
[8,2,2] => 1
[8,2,1,1] => 2
[8,1,1,1,1] => 1
[7,5] => 1
[7,4,1] => 2
[7,3,2] => 2
[7,3,1,1] => 2
[7,2,2,1] => 1
[7,2,1,1,1] => 2
[7,1,1,1,1,1] => 1
[6,6] => 1
[6,5,1] => 2
[6,4,2] => 2
[6,4,1,1] => 2
[6,3,3] => 1
[6,3,2,1] => 3
[6,3,1,1,1] => 2
[6,2,2,2] => 1
[6,2,2,1,1] => 2
[6,2,1,1,1,1] => 2
[6,1,1,1,1,1,1] => 1
[5,5,2] => 1
[5,5,1,1] => 2
[5,4,3] => 2
[5,4,2,1] => 3
[5,4,1,1,1] => 2
[5,3,3,1] => 1
[5,3,2,2] => 2
[5,3,2,1,1] => 3
[5,3,1,1,1,1] => 2
[5,2,2,2,1] => 1
[5,2,2,1,1,1] => 2
[5,2,1,1,1,1,1] => 2
[5,1,1,1,1,1,1,1] => 1
[4,4,4] => 1
[4,4,3,1] => 2
[4,4,2,2] => 2
[4,4,2,1,1] => 2
[4,4,1,1,1,1] => 2
[4,3,3,2] => 1
[4,3,3,1,1] => 2
[4,3,2,2,1] => 2
[4,3,2,1,1,1] => 3
[4,3,1,1,1,1,1] => 2
[4,2,2,2,2] => 1
[4,2,2,2,1,1] => 2
[4,2,2,1,1,1,1] => 2
[4,2,1,1,1,1,1,1] => 2
[4,1,1,1,1,1,1,1,1] => 1
[3,3,3,3] => 1
[3,3,3,2,1] => 2
[3,3,3,1,1,1] => 2
[3,3,2,2,2] => 2
[3,3,2,2,1,1] => 3
[3,3,2,1,1,1,1] => 2
[3,3,1,1,1,1,1,1] => 2
[3,2,2,2,2,1] => 1
[3,2,2,2,1,1,1] => 2
[3,2,2,1,1,1,1,1] => 2
[3,2,1,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2] => 1
[2,2,2,2,2,1,1] => 2
[2,2,2,2,1,1,1,1] => 2
[2,2,2,1,1,1,1,1,1] => 2
[2,2,1,1,1,1,1,1,1,1] => 2
[2,1,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
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Description
The number of upper covers of a partition in dominance order.
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.upper_covers(Q(pi)))
Created
May 09, 2016 at 10:40 by Christian Stump
Updated
Oct 29, 2017 at 21:36 by Martin Rubey