Finding all possible partitions of N units into k classe

Can you tell us which package that function is in?
Google on the r-project site nor on the www produced a hit.
best, ingmar

From: "Ales Ziberna" <aleszib at gmail.com>
Date: Fri, 9 Dec 2005 09:22:47 +0100
To: "R-help" <r-help at stat.math.ethz.ch>
Subject: Re: [R] Finding all possible partitions of N units into k
classe

I would like to thank everybody who replied for their useful
suggestions and
especially the person who (since you replied privately, I do not know
if I
may expose your name or function) provided the "nkpartitions"
function, that
does exactly what I wanted.



Thank you all again!


Best,

Ales Ziberna

----- Original Message -----
From: "Ted Harding" <Ted.Harding at nessie.mcc.ac.uk>
To: "Ales Ziberna" <aleszib at gmail.com>
Cc: "R-help" <r-help at stat.math.ethz.ch>
Sent: Thursday, December 08, 2005 5:19 PM
Subject: RE: [R] Finding all possible partitions of N units into k
classe


On 08-Dec-05 Ales Ziberna wrote:

Dear useRs!

I would like to generate a list of all possible (unique)
partitions of N units into k classes. For example, all possible
partitions of 4 units into 2 classes are (I hope I have not
missed anyone):

1,1,1,2 (this can be read as {1,2,3},{4})
1,1,2,1
1,2,1,1
2,1,1,1
1,1,2,2
1,2,1,2
1,2,2,1

The partitions 1,1,2,2 and 2,2,1,1 are the same and are
therefore not two unique partitions.

... which seems to imply that 2,1,1,1 and 1,2,2,2 are the same,
so I would write your list above as

1,1,1,2 (this can be read as {1,2,3},{4})
1,1,2,1
1,2,1,1
1,2,2,2
1,1,2,2
1,2,1,2
1,2,2,1

which should be a clue!

Fix the class to which unit "1" belongs as Class 1. This
leaves the partitioning of units 2:N, of which there are
2^(N-1) except that you want to exclude the case where they
all go into Class 1. So 2^(N-1) -1.

So let K = 1:(2^(N-1)-1), and for each k in K make the binary
representation of k. Say this gives N-1 binary digits

i1 i2 ... i[N-1]

(note that none of these will have all binary digits = 0).

Then assign unit "j+1" to Class 1 if ij = 0, otherwise to
Class 2.

However, that is if you want to do it with your bare hands!
The package combinat contains also the function 'hcube' which
can be readily adapted to do just that (since it initially
generates all the 2^N combinations of the above).

library(combinat)
?hcube

x<-rep(2,4) # for partitions of 4 units into classes {1,2}

hcube(x,scale=1,transl=0)
#       [,1] [,2] [,3] [,4]
#  [1,]    1    1    1    1
#  [2,]    2    1    1    1
#  [3,]    1    2    1    1
#  [4,]    2    2    1    1
#  [5,]    1    1    2    1
#  [6,]    2    1    2    1
#  [7,]    1    2    2    1
#  [8,]    2    2    2    1
#  [9,]    1    1    1    2
# [10,]    2    1    1    2
# [11,]    1    2    1    2
# [12,]    2    2    1    2
# [13,]    1    1    2    2
# [14,]    2    1    2    2
# [15,]    1    2    2    2
# [16,]    2    2    2    2

### Note, by following the "2"s, that this is counting in binary
### from 0 to 2^N - 1, with "1" for 0 and "2" for 1 and least
### significant bit on the left, so it does what is described
### above. But we need to manipulate this, so assign it to K:

K<-hcube(x,scale=1,transl=0)

### Now select only thos which assign unit "1" to Class 1:

K[K[,1]==1,]
#      [,1] [,2] [,3] [,4]
# [1,]    1    1    1    1
# [2,]    1    2    1    1
# [3,]    1    1    2    1
# [4,]    1    2    2    1
# [5,]    1    1    1    2
# [6,]    1    2    1    2
# [7,]    1    1    2    2
# [8,]    1    2    2    2

of which you need to leave off the first, so, finally:

N<-4  ### Or general N at this point

x<-rep(2,N)

K<-hcube(x,scale=1,transl=0)

K[K[,1]==1,][-1,]
#      [,1] [,2] [,3] [,4]
# [1,]    1    2    1    1
# [2,]    1    1    2    1
# [3,]    1    2    2    1
# [4,]    1    1    1    2
# [5,]    1    2    1    2
# [6,]    1    1    2    2
# [7,]    1    2    2    2


That looks like it!

Best wishes,
Ted.


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Date: 08-Dec-05                                       Time: 16:19:24
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