The partitions
package provides efficient vectorized code to enumerate
solutions to various integer equations. For example, we might note that
and we might want to list all seven in a consistent format (note here that each sum is written in nonincreasing order, so is considered to be the same as ).
You can install the released version of wedge from CRAN with:
# install.packages("partitions") # uncomment this to install the package
library("partitions")
partitions
package in useTo enumerate the partitions of 5:
parts(5)
#>
#> [1,] 5 4 3 3 2 2 1
#> [2,] 0 1 2 1 2 1 1
#> [3,] 0 0 0 1 1 1 1
#> [4,] 0 0 0 0 0 1 1
#> [5,] 0 0 0 0 0 0 1
(each column is padded with zeros). Of course, larger integers have many
more partitions and in this case we can use summary()
:
summary(parts(16))
#>
#> [1,] 16 15 14 14 13 13 13 12 12 12 ... 3 2 2 2 2 2 2 2 2 1
#> [2,] 0 1 2 1 3 2 1 4 3 2 ... 1 2 2 2 2 2 2 2 1 1
#> [3,] 0 0 0 1 0 1 1 0 1 2 ... 1 2 2 2 2 2 2 1 1 1
#> [4,] 0 0 0 0 0 0 1 0 0 0 ... 1 2 2 2 2 2 1 1 1 1
#> [5,] 0 0 0 0 0 0 0 0 0 0 ... 1 2 2 2 2 1 1 1 1 1
#> [6,] 0 0 0 0 0 0 0 0 0 0 ... 1 2 2 2 1 1 1 1 1 1
#> [7,] 0 0 0 0 0 0 0 0 0 0 ... 1 2 2 1 1 1 1 1 1 1
#> [8,] 0 0 0 0 0 0 0 0 0 0 ... 1 2 1 1 1 1 1 1 1 1
#> [9,] 0 0 0 0 0 0 0 0 0 0 ... 1 0 1 1 1 1 1 1 1 1
#> [10,] 0 0 0 0 0 0 0 0 0 0 ... 1 0 0 1 1 1 1 1 1 1
#> [11,] 0 0 0 0 0 0 0 0 0 0 ... 1 0 0 0 1 1 1 1 1 1
#> [12,] 0 0 0 0 0 0 0 0 0 0 ... 1 0 0 0 0 1 1 1 1 1
#> [13,] 0 0 0 0 0 0 0 0 0 0 ... 1 0 0 0 0 0 1 1 1 1
#> [14,] 0 0 0 0 0 0 0 0 0 0 ... 1 0 0 0 0 0 0 1 1 1
#> [15,] 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 1 1
#> [16,] 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 1
Sometimes we want to find the unequal partitions (that is, partitions without repeats):
summary(diffparts(16))
#>
#> [1,] 16 15 14 13 13 12 12 11 11 11 ... 8 8 7 7 7 7 7 6 6 6
#> [2,] 0 1 2 3 2 4 3 5 4 3 ... 5 4 6 6 5 5 4 5 5 4
#> [3,] 0 0 0 0 1 0 1 0 1 2 ... 2 3 3 2 4 3 3 4 3 3
#> [4,] 0 0 0 0 0 0 0 0 0 0 ... 1 1 0 1 0 1 2 1 2 2
#> [5,] 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 1
Sometimes we have restrictions on the partition. For example, to
enumerate the partitions of 9 into 5 parts we would use
restrictedparts()
:
summary(restrictedparts(9,5))
#>
#> [1,] 9 8 7 6 5 7 6 5 4 5 ... 5 4 4 3 3 5 4 3 3 2
#> [2,] 0 1 2 3 4 1 2 3 4 2 ... 2 3 2 3 2 1 2 3 2 2
#> [3,] 0 0 0 0 0 1 1 1 1 2 ... 1 1 2 2 2 1 1 1 2 2
#> [4,] 0 0 0 0 0 0 0 0 0 0 ... 1 1 1 1 2 1 1 1 1 2
#> [5,] 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 1 1 1 1 1
and if we want the partitions of 9 into parts not exceeding 5 we would use the conjugate of this:
summary(conjugate(restrictedparts(9,5)))
#>
#> [1,] 1 2 2 2 2 3 3 3 3 3 ... 4 4 4 4 4 5 5 5 5 5
#> [2,] 1 1 2 2 2 1 2 2 2 3 ... 2 2 3 3 4 1 2 2 3 4
#> [3,] 1 1 1 2 2 1 1 2 2 1 ... 1 2 1 2 1 1 1 2 1 0
#> [4,] 1 1 1 1 2 1 1 1 2 1 ... 1 1 1 0 0 1 1 0 0 0
#> [5,] 1 1 1 1 1 1 1 1 0 1 ... 1 0 0 0 0 1 0 0 0 0
#> [6,] 1 1 1 1 0 1 1 0 0 0 ... 0 0 0 0 0 0 0 0 0 0
#> [7,] 1 1 1 0 0 1 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0
#> [8,] 1 1 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0
#> [9,] 1 0 0 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0
Sometimes we have restrictions on each element of a partition and in
this case we would use blockparts()
:
summary(blockparts(1:6,10))
#>
#> [1,] 1 1 1 1 0 1 1 1 0 1 ... 0 1 0 0 0 1 0 0 0 0
#> [2,] 2 2 2 1 2 2 2 1 2 2 ... 0 0 1 0 0 0 1 0 0 0
#> [3,] 3 3 2 3 3 3 2 3 3 1 ... 2 0 0 1 0 0 0 1 0 0
#> [4,] 4 3 4 4 4 2 3 3 3 4 ... 0 1 1 1 2 0 0 0 1 0
#> [5,] 0 1 1 1 1 2 2 2 2 2 ... 2 2 2 2 2 3 3 3 3 4
#> [6,] 0 0 0 0 0 0 0 0 0 0 ... 6 6 6 6 6 6 6 6 6 6
which would show all solutions to , .
Above we considered and to be the same partition, but if these are considered to be distinct, we need the compositions, not partitions:
compositions(4)
#>
#> [1,] 4 1 2 1 3 1 2 1
#> [2,] 0 3 2 1 1 2 1 1
#> [3,] 0 0 0 2 0 1 1 1
#> [4,] 0 0 0 0 0 0 0 1
A set of 4 elements, WLOG
,
may be partitioned into subsets in a number of ways and these are
enumerated with the setparts()
function:
setparts(4)
#>
#> [1,] 1 1 1 1 2 1 1 1 1 1 1 2 2 2 1
#> [2,] 1 1 1 2 1 2 1 2 2 1 2 1 1 3 2
#> [3,] 1 2 1 1 1 2 2 1 3 2 1 3 1 1 3
#> [4,] 1 1 2 1 1 1 2 2 1 3 3 1 3 1 4
In the above, column 2 3 1 1
would correspond to the set partition
.
Knuth deals with multisets (that is, a generalization of the concept of set, in which elements may appear more than once) and gives an algorithm for enumerating a multiset. His simplest example is the permutations of :
multiset(c(1,2,2,3))
#>
#> [1,] 1 1 1 2 2 2 2 2 2 3 3 3
#> [2,] 2 2 3 1 1 2 2 3 3 1 2 2
#> [3,] 2 3 2 2 3 1 3 1 2 2 1 2
#> [4,] 3 2 2 3 2 3 1 2 1 2 2 1
It is possible to answer questions such as the permutations of the word “pepper”:
library("magrittr")
"pepper" %>%
strsplit("") %>%
unlist %>%
match(letters) %>%
multiset %>%
apply(2,function(x){x %>% `[`(letters,.) %>% paste(collapse="")})
#> [1] "eepppr" "eepprp" "eeprpp" "eerppp" "epeppr" "epeprp" "eperpp" "eppepr"
#> [9] "epperp" "eppper" "epppre" "epprep" "epprpe" "eprepp" "eprpep" "eprppe"
#> [17] "ereppp" "erpepp" "erppep" "erpppe" "peeppr" "peeprp" "peerpp" "pepepr"
#> [25] "peperp" "pepper" "peppre" "peprep" "peprpe" "perepp" "perpep" "perppe"
#> [33] "ppeepr" "ppeerp" "ppeper" "ppepre" "pperep" "pperpe" "pppeer" "pppere"
#> [41] "pppree" "ppreep" "pprepe" "pprpee" "preepp" "prepep" "preppe" "prpeep"
#> [49] "prpepe" "prppee" "reeppp" "repepp" "reppep" "repppe" "rpeepp" "rpepep"
#> [57] "rpeppe" "rppeep" "rppepe" "rpppee"
A
riffle shuffle is an ordering
of
integers
such that
and
appear in their original order: if
,
then , and if
,
then . The two groups of integers appear in their original
order. To enumerate all
riffles, use riffle()
:
riffle(2,4)
#>
#> [1,] 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3
#> [2,] 2 3 3 3 3 1 1 1 1 4 4 4 4 4 4
#> [3,] 3 2 4 4 4 2 4 4 4 1 1 1 5 5 5
#> [4,] 4 4 2 5 5 4 2 5 5 2 5 5 1 1 6
#> [5,] 5 5 5 2 6 5 5 2 6 5 2 6 2 6 1
#> [6,] 6 6 6 6 2 6 6 6 2 6 6 2 6 2 2
To enumerate all riffles with sizes
,
use genrif()
:
genrif(1:3)
#>
#> [1,] 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4
#> [2,] 2 2 2 2 4 4 4 4 4 4 1 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 1 1 1 1 1 1 2
#> [3,] 3 4 4 4 2 2 2 5 5 5 3 4 4 4 1 4 4 4 1 1 1 3 3 3 5 5 5 5 5 5 2 2 2 5 5 5 1
#> [4,] 4 3 5 5 3 5 5 2 2 6 4 3 5 5 4 1 5 5 3 5 5 1 5 5 1 1 3 3 6 6 3 5 5 2 2 6 3
#> [5,] 5 5 3 6 5 3 6 3 6 2 5 5 3 6 5 5 1 6 5 3 6 5 1 6 3 6 1 6 1 3 5 3 6 3 6 2 5
#> [6,] 6 6 6 3 6 6 3 6 3 3 6 6 6 3 6 6 6 1 6 6 3 6 6 1 6 3 6 1 3 1 6 6 3 6 3 3 6
#>
#> [1,] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
#> [2,] 2 2 2 2 2 2 2 2 2 2 2 5 5 5 5 5 5 5 5 5 5 5 5
#> [3,] 1 1 3 3 3 5 5 5 5 5 5 1 1 1 2 2 2 2 2 2 6 6 6
#> [4,] 5 5 1 5 5 1 1 3 3 6 6 2 2 6 1 1 3 3 6 6 1 2 2
#> [5,] 3 6 5 1 6 3 6 1 6 1 3 3 6 2 3 6 1 6 1 3 2 1 3
#> [6,] 6 3 6 6 1 6 3 6 1 3 1 6 3 3 6 3 6 1 3 1 3 3 1
For more detail, see the package vignettes
vignette("partitionspaper")
vignette("setpartitions")
vignette("scrabble")