Identifier
Identifier
Values
[] generating graphics... => 1
[1] generating graphics... => 1
[2] generating graphics... => 1
[1,1] generating graphics... => 1
[3] generating graphics... => 1
[2,1] generating graphics... => 2
[1,1,1] generating graphics... => 1
[4] generating graphics... => 1
[3,1] generating graphics... => 2
[2,2] generating graphics... => 1
[2,1,1] generating graphics... => 3
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 1
[4,1] generating graphics... => 2
[3,2] generating graphics... => 2
[3,1,1] generating graphics... => 3
[2,2,1] generating graphics... => 3
[2,1,1,1] generating graphics... => 4
[1,1,1,1,1] generating graphics... => 1
[6] generating graphics... => 1
[5,1] generating graphics... => 2
[4,2] generating graphics... => 2
[4,1,1] generating graphics... => 3
[3,3] generating graphics... => 1
[3,2,1] generating graphics... => 6
[3,1,1,1] generating graphics... => 4
[2,2,2] generating graphics... => 1
[2,2,1,1] generating graphics... => 6
[2,1,1,1,1] generating graphics... => 5
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 1
[6,1] generating graphics... => 2
[5,2] generating graphics... => 2
[5,1,1] generating graphics... => 3
[4,3] generating graphics... => 2
[4,2,1] generating graphics... => 6
[4,1,1,1] generating graphics... => 4
[3,3,1] generating graphics... => 3
[3,2,2] generating graphics... => 3
[3,2,1,1] generating graphics... => 12
[3,1,1,1,1] generating graphics... => 5
[2,2,2,1] generating graphics... => 4
[2,2,1,1,1] generating graphics... => 10
[2,1,1,1,1,1] generating graphics... => 6
[1,1,1,1,1,1,1] generating graphics... => 1
[8] generating graphics... => 1
[7,1] generating graphics... => 2
[6,2] generating graphics... => 2
[6,1,1] generating graphics... => 3
[5,3] generating graphics... => 2
[5,2,1] generating graphics... => 6
[5,1,1,1] generating graphics... => 4
[4,4] generating graphics... => 1
[4,3,1] generating graphics... => 6
[4,2,2] generating graphics... => 3
[4,2,1,1] generating graphics... => 12
[4,1,1,1,1] generating graphics... => 5
[3,3,2] generating graphics... => 3
[3,3,1,1] generating graphics... => 6
[3,2,2,1] generating graphics... => 12
[3,2,1,1,1] generating graphics... => 20
[3,1,1,1,1,1] generating graphics... => 6
[2,2,2,2] generating graphics... => 1
[2,2,2,1,1] generating graphics... => 10
[2,2,1,1,1,1] generating graphics... => 15
[2,1,1,1,1,1,1] generating graphics... => 7
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 1
[8,1] generating graphics... => 2
[7,2] generating graphics... => 2
[7,1,1] generating graphics... => 3
[6,3] generating graphics... => 2
[6,2,1] generating graphics... => 6
[6,1,1,1] generating graphics... => 4
[5,4] generating graphics... => 2
[5,3,1] generating graphics... => 6
[5,2,2] generating graphics... => 3
[5,2,1,1] generating graphics... => 12
[5,1,1,1,1] generating graphics... => 5
[4,4,1] generating graphics... => 3
[4,3,2] generating graphics... => 6
[4,3,1,1] generating graphics... => 12
[4,2,2,1] generating graphics... => 12
[4,2,1,1,1] generating graphics... => 20
[4,1,1,1,1,1] generating graphics... => 6
[3,3,3] generating graphics... => 1
[3,3,2,1] generating graphics... => 12
[3,3,1,1,1] generating graphics... => 10
[3,2,2,2] generating graphics... => 4
[3,2,2,1,1] generating graphics... => 30
[3,2,1,1,1,1] generating graphics... => 30
[3,1,1,1,1,1,1] generating graphics... => 7
[2,2,2,2,1] generating graphics... => 5
[2,2,2,1,1,1] generating graphics... => 20
[2,2,1,1,1,1,1] generating graphics... => 21
[2,1,1,1,1,1,1,1] generating graphics... => 8
[1,1,1,1,1,1,1,1,1] generating graphics... => 1
[10] generating graphics... => 1
[9,1] generating graphics... => 2
[8,2] generating graphics... => 2
[8,1,1] generating graphics... => 3
[7,3] generating graphics... => 2
[7,2,1] generating graphics... => 6
[7,1,1,1] generating graphics... => 4
[6,4] generating graphics... => 2
[6,3,1] generating graphics... => 6
[6,2,2] generating graphics... => 3
[6,2,1,1] generating graphics... => 12
[6,1,1,1,1] generating graphics... => 5
[5,5] generating graphics... => 1
[5,4,1] generating graphics... => 6
[5,3,2] generating graphics... => 6
[5,3,1,1] generating graphics... => 12
[5,2,2,1] generating graphics... => 12
[5,2,1,1,1] generating graphics... => 20
[5,1,1,1,1,1] generating graphics... => 6
[4,4,2] generating graphics... => 3
[4,4,1,1] generating graphics... => 6
[4,3,3] generating graphics... => 3
[4,3,2,1] generating graphics... => 24
[4,3,1,1,1] generating graphics... => 20
[4,2,2,2] generating graphics... => 4
[4,2,2,1,1] generating graphics... => 30
[4,2,1,1,1,1] generating graphics... => 30
[4,1,1,1,1,1,1] generating graphics... => 7
[3,3,3,1] generating graphics... => 4
[3,3,2,2] generating graphics... => 6
[3,3,2,1,1] generating graphics... => 30
[3,3,1,1,1,1] generating graphics... => 15
[3,2,2,2,1] generating graphics... => 20
[3,2,2,1,1,1] generating graphics... => 60
[3,2,1,1,1,1,1] generating graphics... => 42
[3,1,1,1,1,1,1,1] generating graphics... => 8
[2,2,2,2,2] generating graphics... => 1
[2,2,2,2,1,1] generating graphics... => 15
[2,2,2,1,1,1,1] generating graphics... => 35
[2,2,1,1,1,1,1,1] generating graphics... => 28
[2,1,1,1,1,1,1,1,1] generating graphics... => 9
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[11] generating graphics... => 1
[10,1] generating graphics... => 2
[9,2] generating graphics... => 2
[9,1,1] generating graphics... => 3
[8,3] generating graphics... => 2
[8,2,1] generating graphics... => 6
[8,1,1,1] generating graphics... => 4
[7,4] generating graphics... => 2
[7,3,1] generating graphics... => 6
[7,2,2] generating graphics... => 3
[7,2,1,1] generating graphics... => 12
[7,1,1,1,1] generating graphics... => 5
[6,5] generating graphics... => 2
[6,4,1] generating graphics... => 6
[6,3,2] generating graphics... => 6
[6,3,1,1] generating graphics... => 12
[6,2,2,1] generating graphics... => 12
[6,2,1,1,1] generating graphics... => 20
[6,1,1,1,1,1] generating graphics... => 6
[5,5,1] generating graphics... => 3
[5,4,2] generating graphics... => 6
[5,4,1,1] generating graphics... => 12
[5,3,3] generating graphics... => 3
[5,3,2,1] generating graphics... => 24
[5,3,1,1,1] generating graphics... => 20
[5,2,2,2] generating graphics... => 4
[5,2,2,1,1] generating graphics... => 30
[5,2,1,1,1,1] generating graphics... => 30
[5,1,1,1,1,1,1] generating graphics... => 7
[4,4,3] generating graphics... => 3
[4,4,2,1] generating graphics... => 12
[4,4,1,1,1] generating graphics... => 10
[4,3,3,1] generating graphics... => 12
[4,3,2,2] generating graphics... => 12
[4,3,2,1,1] generating graphics... => 60
[4,3,1,1,1,1] generating graphics... => 30
[4,2,2,2,1] generating graphics... => 20
[4,2,2,1,1,1] generating graphics... => 60
[4,2,1,1,1,1,1] generating graphics... => 42
[4,1,1,1,1,1,1,1] generating graphics... => 8
[3,3,3,2] generating graphics... => 4
[3,3,3,1,1] generating graphics... => 10
[3,3,2,2,1] generating graphics... => 30
[3,3,2,1,1,1] generating graphics... => 60
[3,3,1,1,1,1,1] generating graphics... => 21
[3,2,2,2,2] generating graphics... => 5
[3,2,2,2,1,1] generating graphics... => 60
[3,2,2,1,1,1,1] generating graphics... => 105
[3,2,1,1,1,1,1,1] generating graphics... => 56
[3,1,1,1,1,1,1,1,1] generating graphics... => 9
[2,2,2,2,2,1] generating graphics... => 6
[2,2,2,2,1,1,1] generating graphics... => 35
[2,2,2,1,1,1,1,1] generating graphics... => 56
[2,2,1,1,1,1,1,1,1] generating graphics... => 36
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 10
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[12] generating graphics... => 1
[11,1] generating graphics... => 2
[10,2] generating graphics... => 2
[10,1,1] generating graphics... => 3
[9,3] generating graphics... => 2
[9,2,1] generating graphics... => 6
[9,1,1,1] generating graphics... => 4
[8,4] generating graphics... => 2
[8,3,1] generating graphics... => 6
[8,2,2] generating graphics... => 3
[8,2,1,1] generating graphics... => 12
[8,1,1,1,1] generating graphics... => 5
[7,5] generating graphics... => 2
[7,4,1] generating graphics... => 6
[7,3,2] generating graphics... => 6
[7,3,1,1] generating graphics... => 12
[7,2,2,1] generating graphics... => 12
[7,2,1,1,1] generating graphics... => 20
[7,1,1,1,1,1] generating graphics... => 6
[6,6] generating graphics... => 1
[6,5,1] generating graphics... => 6
[6,4,2] generating graphics... => 6
[6,4,1,1] generating graphics... => 12
[6,3,3] generating graphics... => 3
[6,3,2,1] generating graphics... => 24
[6,3,1,1,1] generating graphics... => 20
[6,2,2,2] generating graphics... => 4
[6,2,2,1,1] generating graphics... => 30
[6,2,1,1,1,1] generating graphics... => 30
[6,1,1,1,1,1,1] generating graphics... => 7
[5,5,2] generating graphics... => 3
[5,5,1,1] generating graphics... => 6
[5,4,3] generating graphics... => 6
[5,4,2,1] generating graphics... => 24
[5,4,1,1,1] generating graphics... => 20
[5,3,3,1] generating graphics... => 12
[5,3,2,2] generating graphics... => 12
[5,3,2,1,1] generating graphics... => 60
[5,3,1,1,1,1] generating graphics... => 30
[5,2,2,2,1] generating graphics... => 20
[5,2,2,1,1,1] generating graphics... => 60
[5,2,1,1,1,1,1] generating graphics... => 42
[5,1,1,1,1,1,1,1] generating graphics... => 8
[4,4,4] generating graphics... => 1
[4,4,3,1] generating graphics... => 12
[4,4,2,2] generating graphics... => 6
[4,4,2,1,1] generating graphics... => 30
[4,4,1,1,1,1] generating graphics... => 15
[4,3,3,2] generating graphics... => 12
[4,3,3,1,1] generating graphics... => 30
[4,3,2,2,1] generating graphics... => 60
[4,3,2,1,1,1] generating graphics... => 120
[4,3,1,1,1,1,1] generating graphics... => 42
[4,2,2,2,2] generating graphics... => 5
[4,2,2,2,1,1] generating graphics... => 60
[4,2,2,1,1,1,1] generating graphics... => 105
[4,2,1,1,1,1,1,1] generating graphics... => 56
[4,1,1,1,1,1,1,1,1] generating graphics... => 9
[3,3,3,3] generating graphics... => 1
[3,3,3,2,1] generating graphics... => 20
[3,3,3,1,1,1] generating graphics... => 20
[3,3,2,2,2] generating graphics... => 10
[3,3,2,2,1,1] generating graphics... => 90
[3,3,2,1,1,1,1] generating graphics... => 105
[3,3,1,1,1,1,1,1] generating graphics... => 28
[3,2,2,2,2,1] generating graphics... => 30
[3,2,2,2,1,1,1] generating graphics... => 140
[3,2,2,1,1,1,1,1] generating graphics... => 168
[3,2,1,1,1,1,1,1,1] generating graphics... => 72
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 10
[2,2,2,2,2,2] generating graphics... => 1
[2,2,2,2,2,1,1] generating graphics... => 21
[2,2,2,2,1,1,1,1] generating graphics... => 70
[2,2,2,1,1,1,1,1,1] generating graphics... => 84
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 45
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 11
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
click to show generating function       
Description
The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions.
This is the multinomial of the multiplicities of the parts, see [1].
References
[1] Preferred multisets: triangle of numbers refining A007318 using format described in A036038. OEIS:A048996
Code
def statistic(la):
    return multinomial(la.to_exp())

#CodeLanguage: Sage
def to_partition(elt):
    from sage.combinat.partition import Partition
    return Partition(sorted(elt, reverse=True))

@cached_function
def preimages(level):
    print "computing preimages for level", level
    result = dict()
    for el in Compositions(level):
        image = to_partition(el)
        result[image] = result.get(image, 0) + 1
    return result

def statistic(x):
    return preimages(x.size()).get(x, 0)

Created
Sep 11, 2015 at 22:04 by Martin Rubey
Updated
Oct 29, 2017 at 20:49 by Martin Rubey