Identifier
Values
=>
Cc0002;cc-rep
[]=>1
[1]=>1
[2]=>1
[1,1]=>1
[3]=>1
[2,1]=>2
[1,1,1]=>1
[4]=>1
[3,1]=>2
[2,2]=>1
[2,1,1]=>3
[1,1,1,1]=>1
[5]=>1
[4,1]=>2
[3,2]=>2
[3,1,1]=>3
[2,2,1]=>3
[2,1,1,1]=>4
[1,1,1,1,1]=>1
[6]=>1
[5,1]=>2
[4,2]=>2
[4,1,1]=>3
[3,3]=>1
[3,2,1]=>6
[3,1,1,1]=>4
[2,2,2]=>1
[2,2,1,1]=>6
[2,1,1,1,1]=>5
[1,1,1,1,1,1]=>1
[7]=>1
[6,1]=>2
[5,2]=>2
[5,1,1]=>3
[4,3]=>2
[4,2,1]=>6
[4,1,1,1]=>4
[3,3,1]=>3
[3,2,2]=>3
[3,2,1,1]=>12
[3,1,1,1,1]=>5
[2,2,2,1]=>4
[2,2,1,1,1]=>10
[2,1,1,1,1,1]=>6
[1,1,1,1,1,1,1]=>1
[8]=>1
[7,1]=>2
[6,2]=>2
[6,1,1]=>3
[5,3]=>2
[5,2,1]=>6
[5,1,1,1]=>4
[4,4]=>1
[4,3,1]=>6
[4,2,2]=>3
[4,2,1,1]=>12
[4,1,1,1,1]=>5
[3,3,2]=>3
[3,3,1,1]=>6
[3,2,2,1]=>12
[3,2,1,1,1]=>20
[3,1,1,1,1,1]=>6
[2,2,2,2]=>1
[2,2,2,1,1]=>10
[2,2,1,1,1,1]=>15
[2,1,1,1,1,1,1]=>7
[1,1,1,1,1,1,1,1]=>1
[9]=>1
[8,1]=>2
[7,2]=>2
[7,1,1]=>3
[6,3]=>2
[6,2,1]=>6
[6,1,1,1]=>4
[5,4]=>2
[5,3,1]=>6
[5,2,2]=>3
[5,2,1,1]=>12
[5,1,1,1,1]=>5
[4,4,1]=>3
[4,3,2]=>6
[4,3,1,1]=>12
[4,2,2,1]=>12
[4,2,1,1,1]=>20
[4,1,1,1,1,1]=>6
[3,3,3]=>1
[3,3,2,1]=>12
[3,3,1,1,1]=>10
[3,2,2,2]=>4
[3,2,2,1,1]=>30
[3,2,1,1,1,1]=>30
[3,1,1,1,1,1,1]=>7
[2,2,2,2,1]=>5
[2,2,2,1,1,1]=>20
[2,2,1,1,1,1,1]=>21
[2,1,1,1,1,1,1,1]=>8
[1,1,1,1,1,1,1,1,1]=>1
[10]=>1
[9,1]=>2
[8,2]=>2
[8,1,1]=>3
[7,3]=>2
[7,2,1]=>6
[7,1,1,1]=>4
[6,4]=>2
[6,3,1]=>6
[6,2,2]=>3
[6,2,1,1]=>12
[6,1,1,1,1]=>5
[5,5]=>1
[5,4,1]=>6
[5,3,2]=>6
[5,3,1,1]=>12
[5,2,2,1]=>12
[5,2,1,1,1]=>20
[5,1,1,1,1,1]=>6
[4,4,2]=>3
[4,4,1,1]=>6
[4,3,3]=>3
[4,3,2,1]=>24
[4,3,1,1,1]=>20
[4,2,2,2]=>4
[4,2,2,1,1]=>30
[4,2,1,1,1,1]=>30
[4,1,1,1,1,1,1]=>7
[3,3,3,1]=>4
[3,3,2,2]=>6
[3,3,2,1,1]=>30
[3,3,1,1,1,1]=>15
[3,2,2,2,1]=>20
[3,2,2,1,1,1]=>60
[3,2,1,1,1,1,1]=>42
[3,1,1,1,1,1,1,1]=>8
[2,2,2,2,2]=>1
[2,2,2,2,1,1]=>15
[2,2,2,1,1,1,1]=>35
[2,2,1,1,1,1,1,1]=>28
[2,1,1,1,1,1,1,1,1]=>9
[1,1,1,1,1,1,1,1,1,1]=>1
[11]=>1
[10,1]=>2
[9,2]=>2
[9,1,1]=>3
[8,3]=>2
[8,2,1]=>6
[8,1,1,1]=>4
[7,4]=>2
[7,3,1]=>6
[7,2,2]=>3
[7,2,1,1]=>12
[7,1,1,1,1]=>5
[6,5]=>2
[6,4,1]=>6
[6,3,2]=>6
[6,3,1,1]=>12
[6,2,2,1]=>12
[6,2,1,1,1]=>20
[6,1,1,1,1,1]=>6
[5,5,1]=>3
[5,4,2]=>6
[5,4,1,1]=>12
[5,3,3]=>3
[5,3,2,1]=>24
[5,3,1,1,1]=>20
[5,2,2,2]=>4
[5,2,2,1,1]=>30
[5,2,1,1,1,1]=>30
[5,1,1,1,1,1,1]=>7
[4,4,3]=>3
[4,4,2,1]=>12
[4,4,1,1,1]=>10
[4,3,3,1]=>12
[4,3,2,2]=>12
[4,3,2,1,1]=>60
[4,3,1,1,1,1]=>30
[4,2,2,2,1]=>20
[4,2,2,1,1,1]=>60
[4,2,1,1,1,1,1]=>42
[4,1,1,1,1,1,1,1]=>8
[3,3,3,2]=>4
[3,3,3,1,1]=>10
[3,3,2,2,1]=>30
[3,3,2,1,1,1]=>60
[3,3,1,1,1,1,1]=>21
[3,2,2,2,2]=>5
[3,2,2,2,1,1]=>60
[3,2,2,1,1,1,1]=>105
[3,2,1,1,1,1,1,1]=>56
[3,1,1,1,1,1,1,1,1]=>9
[2,2,2,2,2,1]=>6
[2,2,2,2,1,1,1]=>35
[2,2,2,1,1,1,1,1]=>56
[2,2,1,1,1,1,1,1,1]=>36
[2,1,1,1,1,1,1,1,1,1]=>10
[1,1,1,1,1,1,1,1,1,1,1]=>1
[12]=>1
[11,1]=>2
[10,2]=>2
[10,1,1]=>3
[9,3]=>2
[9,2,1]=>6
[9,1,1,1]=>4
[8,4]=>2
[8,3,1]=>6
[8,2,2]=>3
[8,2,1,1]=>12
[8,1,1,1,1]=>5
[7,5]=>2
[7,4,1]=>6
[7,3,2]=>6
[7,3,1,1]=>12
[7,2,2,1]=>12
[7,2,1,1,1]=>20
[7,1,1,1,1,1]=>6
[6,6]=>1
[6,5,1]=>6
[6,4,2]=>6
[6,4,1,1]=>12
[6,3,3]=>3
[6,3,2,1]=>24
[6,3,1,1,1]=>20
[6,2,2,2]=>4
[6,2,2,1,1]=>30
[6,2,1,1,1,1]=>30
[6,1,1,1,1,1,1]=>7
[5,5,2]=>3
[5,5,1,1]=>6
[5,4,3]=>6
[5,4,2,1]=>24
[5,4,1,1,1]=>20
[5,3,3,1]=>12
[5,3,2,2]=>12
[5,3,2,1,1]=>60
[5,3,1,1,1,1]=>30
[5,2,2,2,1]=>20
[5,2,2,1,1,1]=>60
[5,2,1,1,1,1,1]=>42
[5,1,1,1,1,1,1,1]=>8
[4,4,4]=>1
[4,4,3,1]=>12
[4,4,2,2]=>6
[4,4,2,1,1]=>30
[4,4,1,1,1,1]=>15
[4,3,3,2]=>12
[4,3,3,1,1]=>30
[4,3,2,2,1]=>60
[4,3,2,1,1,1]=>120
[4,3,1,1,1,1,1]=>42
[4,2,2,2,2]=>5
[4,2,2,2,1,1]=>60
[4,2,2,1,1,1,1]=>105
[4,2,1,1,1,1,1,1]=>56
[4,1,1,1,1,1,1,1,1]=>9
[3,3,3,3]=>1
[3,3,3,2,1]=>20
[3,3,3,1,1,1]=>20
[3,3,2,2,2]=>10
[3,3,2,2,1,1]=>90
[3,3,2,1,1,1,1]=>105
[3,3,1,1,1,1,1,1]=>28
[3,2,2,2,2,1]=>30
[3,2,2,2,1,1,1]=>140
[3,2,2,1,1,1,1,1]=>168
[3,2,1,1,1,1,1,1,1]=>72
[3,1,1,1,1,1,1,1,1,1]=>10
[2,2,2,2,2,2]=>1
[2,2,2,2,2,1,1]=>21
[2,2,2,2,1,1,1,1]=>70
[2,2,2,1,1,1,1,1,1]=>84
[2,2,1,1,1,1,1,1,1,1]=>45
[2,1,1,1,1,1,1,1,1,1,1]=>11
[1,1,1,1,1,1,1,1,1,1,1,1]=>1
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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].
This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$,
where $k$ is the number of parts of $\lambda$.
An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$
where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.
This is the multinomial of the multiplicities of the parts, see [1].
This is the same as $m_\lambda(x_1,\dotsc,x_k)$ evaluated at $x_1=\dotsb=x_k=1$,
where $k$ is the number of parts of $\lambda$.
An explicit formula is $\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$
where $m_i(\lambda)$ is the number of parts of $\lambda$ equal to $i$.
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
Nov 29, 2023 at 14:21 by Martin Rubey
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