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.
Thank you in advance for any suggestions.
Best,
Ales Ziberna
Finding all possible partitions of N units into k classes
7 messages · Ingmar Visser, Ales Ziberna, (Ted Harding)
combinations in the gtools package can be helpfull here, best, ingmar
From: "Ales Ziberna" <aleszib at gmail.com>
Date: Thu, 8 Dec 2005 15:45:37 +0100
To: "R-help" <r-help at stat.math.ethz.ch>
Subject: [R] Finding all possible partitions of N units into k classes
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.
Thank you in advance for any suggestions.
Best,
Ales Ziberna
______________________________________________ R-help at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
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.
--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 08-Dec-05 Time: 16:19:24
------------------------------ XFMail ------------------------------
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.
--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 08-Dec-05 Time: 16:19:24
------------------------------ XFMail ------------------------------
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.
--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 08-Dec-05 Time: 16:19:24
------------------------------ XFMail ------------------------------
______________________________________________ R-help at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
On 09-Dec-05 Ingmar Visser wrote:
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
I would be interested in this too! From the name "nkpartitions" it would seem that this function generates the ways of distributing n items over k classes; and, in the context of Ales Ziberna's original query, two such distributions are considered the same if one can be obtained from the other by permuting classes. This last consideration, in particular, is not trivial! Ted.
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.
--------------------------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 08-Dec-05 Time: 16:19:24
------------------------------ XFMail ------------------------------
______________________________________________ R-help at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
______________________________________________ R-help at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
-------------------------------------------------------------------- E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk> Fax-to-email: +44 (0)870 094 0861 Date: 09-Dec-05 Time: 10:05:13 ------------------------------ XFMail ------------------------------
As to my knowladge, the function is not located in any packages. If you wish, I can contact the author and ask him, if he is willing to post the function on to the list. Best, Ales Ziberna ----- Original Message ----- From: "Ingmar Visser" <I.Visser at uva.nl> To: "Ales Ziberna" <aleszib at gmail.com>; "R-help" <r-help at stat.math.ethz.ch> Sent: Friday, December 09, 2005 9:56 AM Subject: Re: [R] 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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E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk>
Fax-to-email: +44 (0)870 094 0861
Date: 08-Dec-05 Time: 16:19:24
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