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-----------------------------------------------------------------------------
Statistic identifier: St000278
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
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 $i$.
-----------------------------------------------------------------------------
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())
-----------------------------------------------------------------------------
Statistic values:
[] => 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
-----------------------------------------------------------------------------
Created: Sep 11, 2015 at 22:04 by Martin Rubey
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Last Updated: Nov 29, 2023 at 14:21 by Martin Rubey