User:Devon McCormick/HAP AIP
All Integer Partitions: J Programs Compared
Howard A. Peelle 2013 OCT 1
University of Massachusetts Amherst
hapeelle@educ.umass.edu
Abstract
J programs for generating all partitions of an integer are presented and compared.
Whoever wants to go about generating all partitions
not only immerses himself in immense labor,
but also must take pains to keep fully attentive,
so as not to be grossly deceived.
-- Leonhard Euler, __De Partitione Numerorum__ (1750)
Introduction
Historically, there has been great interest in partitions, especially computing the number of partitions of an integer. Relatively recently, algorithms and programs for generating all integer partitions have appeared but have not been expressly compared together in a common language. Here, a dozen or so J programs are presented developmentally and are compared by speed, space, and spread.
Note: Programs are organized generally in chronological order by author of algorithm, with alternative coding to contrast efficiency, but with minimum explanation. If you are just starting to learn J, please consult the JSoftware website [0] and read appropriate tutorials, notably [6] and [7]. If you are only interested in the best programs, skip to Comparisons and see Appendix for their final definitions.
A partition of an integer n is represented here as a list of parts: positive integers whose sum is n.
All integer partitions of n are all the distinct partitions of n with parts in ascending (non-decreasing) or descending (non-increasing) order.
Partitions may be listed in any order but usually are in ascending or descending base value. For example, all partitions of 6 in ascending order, with ascending parts:
6
1 5
2 4
3 3
1 1 4
1 2 3
2 2 2
1 1 1 3
1 1 2 2
1 1 1 1 2
1 1 1 1 1 1
Note: The following utility names will be used often throughout, but their definitions will not be repeated.
ELSE =: `
WHEN =: @.
EACH =: &.>
Skiena’s Algorithm
The first program is a simplified coding of Steven Skiena’s algorithm [1] using double recursion to produce partitions of input integer n that begin with input largest part p. It joins two arrays vertically: The top is the result of p joined horizontally onto partitions of n - p with the new largest part being the smaller of n - p and p; the bottom is the partitions of n with largest part p - 1. In tacit J:
Skiena
Skiena =: Parts ELSE Ones WHEN Under2 NB. (n) Skiena (p)
Ones =: ,:@#
Under2 =: 2 > ]
Parts =: Top , Bottom
Top =: ] ,. - Skiena - <. ]
Bottom =: Skiena <:
For example, partitions of 5 beginning with 3 in a table (padded with 0s):
5 Skiena 3 3 2 0 0 0 3 1 1 0 0 2 2 1 0 0 2 1 1 1 0 1 1 1 1 1
All partitions of 6 with parts in descending order:
6 Skiena 6 NB. Skiena~ 6 6 0 0 0 0 0 5 1 0 0 0 0 4 2 0 0 0 0 4 1 1 0 0 0 3 3 0 0 0 0 3 2 1 0 0 0 3 1 1 1 0 0 2 2 2 0 0 0 2 2 1 1 0 0 2 1 1 1 1 0 1 1 1 1 1 1
This is the shortest program here, but it is inefficient in speed and space.
Knuth’s Algorithms
Donald Knuth proffered two algorithms for generating partitions in [2].
Knuth
Knuth’s algorithm P computes successors of partitions, starting with n and ending with a list of 1s: “If a partition isn’t all 1s, it ends with (x+1) followed by zero or more 1s, for some x ≤ 1; therefore, the next smallest partition in lexicographic order is obtained by replacing the suffix (x+1)1…1 by x…xr for some appropriate remainder r ≤ x. The process is quite efficient if we keep track of the largest subscript q [of partition a] such that a,,q,, not equal 1…” [2 page 37]. Coded explicitly in J, the program below performs classic scalar processing to produce a table of all partitions in descending order:
Knuth =: 3 : 0 NB. Knuth (n) for n>:1
n =. y
all =. i.0,n
label_P1. NB. Initialize
a =. n#0
m =.0 NB. origin 0
label_P2. NB. Store final part
a =. n m}a
q =. m - (n=1)
label_P3. NB. Visit a partition
all =. all,(m+1){.a
if. (q{a)~:2 do. goto_P5. end.
label_P4. NB. Change 2 to 1,1
a =. 1 q}a
q =. q-1
m =. m+1
a =. 1 m}a
goto_P3.
label_P5. NB. Decrease q{a
if. q<0 do. all return. end.
x =. (q{a)-1
a =. x q}a
n =. (m-q)+1
m =. q+1
label_P6. NB. Copy x
if. n<:x do. goto_P2. end.
a =. x m}a
m =. m+1
n =. n-x
goto_P6.
)
For example, all partitions of 6 (same result as 6 Skiena 6):
Knuth 6 6 0 0 0 0 0 5 1 0 0 0 0 4 2 0 0 0 0 4 1 1 0 0 0 3 3 0 0 0 0 3 2 1 0 0 0 3 1 1 1 0 0 2 2 2 0 0 0 2 2 1 1 0 0 2 1 1 1 1 0 1 1 1 1 1 1
The program can be shortened drastically, get the remainder neatly, and handle n = 0 as a bonus:
Knuthx =: 3 : 0 NB. Knuthx (n)
all =. ,:p =. ,y
while. 1 < {.p do. i =. <: p i. 1
x =. - <: i { p
a =. i {. p
s =. y - +/a
p =. a , x +/\ s#1
all =. all , p
end.
)
Look at intermediate steps during a loop of Knuthx 10 to produce the next partition after 4 3 1 1 1:
y =. 10
p =. 4 3 1 1 1
]i =. <: p i. 1
1
]x =. - <: i { p
_2
]a =. i {. p
4
]s =. y - +/a
6
]p =. a , x +/\ s#1
4 2 2 2
Unfortunately, Knuthx uses twice the space and is (increasingly) slower than Knuth.
A tacit translation is even slower and fatter in space:
Knutht =: Parts ^: While ^:_ @ N
N =: ,:@,
While =: 1 < {.@{:
Parts =: , Next@Last
Last =: {: -. 0:
Next =: A , X +/\ S#1:
A =: I {. ]
I =: <:@i. 1:
X =: I -.@{ ]
S =: ] -&(+/) A
Using s=.x+(#p)–i instead of s=.y-+/a for the sum of a suffix and including a condition to change a 2 to 1 1 will improve Knuth's speed – especially for large n with growing percentage of 2s:
Knuth2 =: 3 : 0
all =.,:p=.,y
while. 1 < {.p do. i =. <: p i. 1
x =. i { p
if. x=2
do. p =. 1 i} p,1
else.
a =. i {. p
x =. <:x
s =. x + (#p) - i
p =. a , x Next s
end.
all =. all , p
end.
)
Next =: -@[ +/\ ] # 1:
Better yet, use a sub-program to group partitions by their leading part and append an array of all leading 2s separately, with a final row of 1s:
Knuth2s =: 3 : 0
all =. i.0,n=.y
while. n>1
do. nexts =. n Nexts y-n
all =. all , n ,. nexts
n =. n-1
end.
all,1
)
Nexts =: 3 : 0
:
if. y<2 do. ,:,y return. end.
nexts =. ,:next =. x New y
while. 2 < {.next
do. i =. <: next i. 1
x =. i { next
if. x=2
do. next =. 1 i} next,1
else.
a =. i {. next
x =. <:x
s =. x + (#next) - i
next =. a , x New s
end.
nexts =. nexts , next
end. NB. next is 2…(1)
repeats =. >:i.#next-.1
twos =. Two ^:repeats next
nexts,twos
)
Two =: }. , 1 1"_
This is faster and much slimmer, but longer.
Nevertheless, these revisions cannot compete with Knuth in speed or space.
Hindenburg
Knuth’s algorithm H (attributed to Hindenburg [3]) computes partitions of n with m parts -- that is, m-tuples that sum to n. “The basic idea is that colex order goes from one partition a,,1,, … a,,m,, to the next by finding the smallest j such that a j can be increased without changing a,,j+1,, … a,,m,,.” [2 page 38] The new partition will have j leading parts = a,,j,, + 1 and the same sum iff a,,j,, < a,,1,,–1. Note that this algorithm does not work for 0 or 1 parts. Later, see program AP that produces m-tuples robustly. Assuming n ≥ m ≥ 2, code H in J:
H =: 3 : 0 NB. (n) H (m)
:
n =. x NB. n>:m
m =. y NB. m>:2
ps =. i.0,m
label_H1. NB. Initialize
a =. m#0
a =. (1+n-m) 0} a NB. origin 0
a =. 1 (}.i.m)} a
a =. a , _1
label_H2. NB. Visit partition
ps =. ps , (i.m){a
if. (1{a)>:(0{a)-1 do. goto_H4. end.
label_H3. NB. Tweak 0{a and 1{a
a =. ((0{a)-1) 0}a
a =. ((1{a)+1) 1}a
goto_H2.
label_H4. NB. Find j
j =. 2 NB. origin 0
s =. (0{a)+(1{a)-1
while. (j{a)>:(0{a)-1 do. s =. s + j{a
j =. j+1
end.
label_H5. NB. Increase j{a
if. j=m do. ps return. end.
x =. (j{a)+1
a =. x j} a
j =. j-1
label_H6. NB. Tweak (i.j){a
while. j>0 do. a =. x j}a
s =. s-x
j =. j-1
end.
a =. s 0} a
goto_H2.
)
Examples of partitions of 6 with 2, 3, and 4 parts:
6 H 2
5 1
4 2
3 3
6 H 3
4 1 1
3 2 1
2 2 2
6 H 4
3 1 1 1
2 2 1 1
Notice that the order of all partitions is not the same as in Knuth 6.
By appending such results, H can be used in a supra-program to generate all partitions:
Hindenburg =: 3 : 0 NB. Hindenburg (n)
n =. y
m =. 2
all =. ,.n
while. m<:n do. all =. all , n H m
m =. m+1
end.
all
)
Hindenburg is increasingly slower than Knuth and uses about twice the space, so it will be omitted from further comparisons.
Hui’s Algorithm
part
Roger Hui presented a concise program for all partitions in [4]:
pu =: ] <@:+"1 [ * </\"1@=@] pext =: [: ~. [: /:~&.> ,&.> , ;@:(pu&.>) part =: ,&.>`(pext/)@.(1:<#) @ ($&1)
An example of partitions of 6 in a list of 11 boxes with ascending parts:
part 6 ┌───────────┬─────────┬───────┬───────┬─────┬─────┬───┬─────┬───┬───┬─┐ │1 1 1 1 1 1│1 1 1 1 2│1 1 1 3│1 1 2 2│1 1 4│1 2 3│1 5│2 2 2│2 4│3 3│6│ └───────────┴─────────┴───────┴───────┴─────┴─────┴───┴─────┴───┴───┴─┘
To produce a numeric table (padded with 0s), open the list:
>part 6 1 1 1 1 1 1 1 1 1 1 2 0 1 1 1 3 0 0 1 1 2 2 0 0 1 1 4 0 0 0 1 2 3 0 0 0 1 5 0 0 0 0 2 2 2 0 0 0 2 4 0 0 0 0 3 3 0 0 0 0 6 0 0 0 0 0
With or without 0s, this program is woefully slow and obese in space. It cannot compete here.
Boss’s Algorithm
Boss
R. E. Boss developed an efficient algorithm that sparked interest, as well as some revisions. His program from [5] computes all partitions with descending parts:
init =: (<@<@i.@(1 0"_)) ,~ <"0@(] , (] (- <. >:@]) i.)@<:)
pps1 =: >:@i.@[ <@;@:(([ ,. (>: {."1) # ])&.>) {.
exit =: >@{.@>
Boss =: [: exit [: (],~ pps1)&.>/ init
Example:
Boss 6 1 1 1 1 1 1 2 1 1 1 1 0 2 2 1 1 0 0 2 2 2 0 0 0 3 1 1 1 0 0 3 2 1 0 0 0 3 3 0 0 0 0 4 1 1 0 0 0 4 2 0 0 0 0 5 1 0 0 0 0 6 0 0 0 0 0
This program is very fast but fat.
AP0
Boss’s program can be re-coded more perspicuously:
AP0 =: Exit@Part@Init NB. AP0 (n)
Exit =: >@{.@>
Init =: <"0@Mins , <@Empty
Empty =: <@,:@i.@0:
Mins =: , (<. |.)@}.@i.
Part =: Next EACH/
Next =: <@;@Ps , ]
Ps =: Ns@[ Join EACH {.
Ns =: >:@i.
Join =: [ ,. Select # ]
Select =: >: {."1
10% shorter, AP0 runs at about the same speed as Boss in a little less space.
Hui
Hui [6] recast Boss's program using Power instead of Insert:
part =: 3 : 'final (, new)^:y <<i.1 0'
new =: (-i.)@# <@:(cat&.>) ]
cat =: [ ;@:(,.&.>) -@(<.#) {. ]
final =: ; @: (<@-.&0"1&.>) @ > @ {:
part 6 produces a list of 11 boxes of partitions with descending parts (without 0s).
This program is much slower than Boss and uses 80% more space. It could speed up 33% if 0s in boxes were not removed: N.B. final =: ;@>@{: But that would require more space.
Make it conform to other programs here that return a numeric table:
Hui =: >@part (Hui -: Boss) 6 1
This could be re-coded in smaller modules:
HuiBoss =: Final@Parts
Final =: ;@>@{:
Inputs =: ] ` Start
Start =: <@<@,:@i.@0:
Parts =: Boxes^:Inputs
Boxes =: , <@New
New =: (-i.)@# ;@Cat EACH ]
Cat =: [ ,.EACH Min {. ]
Min =: -@<. #
However, these programs need excessive memory and will not compete well here at high n.
AP1
Here is my revision of Boss's program, notably without using an outer level of boxing:
AP1 =: ;@All ELSE Ns WHEN (< 2:) NB. AP1 (n)
All =: Ns ,.EACH Parts/@Mins
Ns =: >:@i.
Mins =: Smaller@}.@i. , 0:
Smaller =: <. |.
Parts =: Next ELSE Start WHEN Zero
Zero =: 0 e. ]
Start =: 1 ; ,@0:
Next =: ;@New ; ]
New =: Ns@[ Join EACH {.
Join =: [ ,. Select # ]
Select =: >: {."1
(AP1 -: Boss) 6
1
This program runs much faster than Boss in significantly less space.
Indeed, it can do n = 70 whereas Boss runs out of memory.
Also consider an explicit translation:
AP1x =: 3 : 0
ns =. >:i.y
mins =. (<. |.) }:ns
all =. < i.1 0
while. #mins do. new =. all New~ {:mins
all =. all ;~ ;new
mins =. }:mins
end.
all =. ns ,.EACH all
;all
)
Notice use of New and its tacit sub-programs from above.
AP1x is slower than AP1, needs a lot more space, and is longer. So it will bow out of the competition.
How about a recursive definition?
AP1r =: ;@All
All =: Ns ,.EACH Mins Parts Empty
Empty =: <@,:@i.@0:
Ns =: >:@i.
Mins =: Smaller@}.@i.
Smaller =: <. |.
Parts =: Build ELSE ] WHEN Done NB. Parts calls Build
Done =: 1 > #@[
Build =: ButFirst Parts First Next ] NB. Build calls Parts
First =: {.@[
ButFirst =: }.@[
Next =: ;@New ; ]
New =: Ns@[ Join EACH {.
Join =: [ ,. Select # ]
Select =: >: {."1
This has about the same speed and space as AP1 but is longer. So, it will defer to AP1.
See Peelle’s Algorithms later for more efficient recursive programs.
AP2
In [7], I described a variant of Boss’s algorithm that computes the leading part at each iteration: the smaller of the number of accumulated boxed arrays and n minus that number. Renamed and edited slightly:
AP2 =: ;@All NB. AP2 (n)
All =: Ns ,.EACH Parts
Ns =: >:@i.
Parts =: Build^:N Start
Start =: <@,:@i.@0:
N =: 0 >. <:@[
Build =: <@;@Next , ]
Next =: Lead Ps ]
Lead =: Min #
Min =: - <. ]
Ps =: Ns@[ Join EACH {.
Join =: [ ,. Select # ]
Select =: >: {."1
See [7] for a tutorial. Example:
AP2 6 1 1 1 1 1 1 2 1 1 1 1 0 2 2 1 1 0 0 2 2 2 0 0 0 3 1 1 1 0 0 3 2 1 0 0 0 3 3 0 0 0 0 4 1 1 0 0 0 4 2 0 0 0 0 5 1 0 0 0 0 6 0 0 0 0 0
This program is about 33% faster than Boss at n= 65 in about 25% less space.
Compared to AP1, it has about the same speed, slightly less space, and less length.
Efficiency can be improved slightly further by separating top and bottom halves of the partitions array, by defining separate programs for odd and even input, or with other programming techniques (not shown here because those programs are much longer).
Kelleher’s Algorithm
Jerome Kelleher and Barry O’Sullivan published two algorithms in [8] for all partitions with ascending parts, superseding speed of existing programs for descending parts by Knuth [2] and by Zoghbi & Stojmenovic [9]. Kelleher’s most efficient algorithm generates lexicographic successors iteratively with embedded loops, notably a second loop to handle transitions involving only the last two parts. Coded straightforwardly in J:
Kelleher
Kelleher =: 3 : 0 NB. Kelleher (n) for n>0
n =. y
a =. (n+1)#0
k =. 1
a =. (0) 0} a NB. redundant
y =. n-1
all =. i.0,n
while. k~:0 do.
x =. ((k-1){a)+1
k =. k-1
while. (2*x)<:y do.
a =. x k} a
y =. y-x
k =. k+1
end.
l =. k+1
while. x<:y do.
a =. x k}a
a =. y l}a
all =. all , (i.k+2){a
x =. x+1
y =. y-1
end.
a =. (x+y) k}a
y =. x + y - 1
all =. all , (i.k+1){a
end.
all
)
Example:
Kelleher 6 1 1 1 1 1 1 1 1 1 1 2 0 1 1 1 3 0 0 1 1 2 2 0 0 1 1 4 0 0 0 1 2 3 0 0 0 1 5 0 0 0 0 2 2 2 0 0 0 2 4 0 0 0 0 3 3 0 0 0 0 6 0 0 0 0 0
Kelleher is 4 times slower than AP2 at n= 65 but uses only about 40% of the space.
It is faster than Knuth in about the same space, and much shorter.
An even faster and shorter, more J-ish version Kelleherx is given in the Appendix.
Peelle’s Algorithms
Now look at how well some new recursive algorithms perform.
AP
Peelle presented a tidy ambivalent program in [7] (edited slightly here) using a “1 Plus” recursive approach to produce all partitions of n or partitions of n with up to p parts:
AP =: ;@All NB. AP (n) or (n) AP (p)
All =: Parts EACH >:@i.
Parts =: Ones ELSE Plus1 WHEN >
Ones =: = # ] ,:@# 1:
Plus1 =: 1 + - AP ]
The main idea is to get partitions with p parts by adding 1 to the sub-array of partitions of n - p (including 0s). In other words, for a given p, distribute n - p into p piles, then add 1 to each pile.
For example, assemble 1 plus each result below to get 6 AP 3:
5 AP 1 6 AP 3
5 6 0 0
4 AP 2 5 1 0
4 0 4 2 0
3 1 3 3 0
2 2 4 1 1
3 AP 3 3 2 1
3 0 0 2 2 2
2 1 0
1 1 1
See [7] for a tutorial. Notice that the order of all partitions is the same as Hindenburg:
AP 6 NB. 6 AP 6 6 0 0 0 0 0 5 1 0 0 0 0 4 2 0 0 0 0 3 3 0 0 0 0 4 1 1 0 0 0 3 2 1 0 0 0 2 2 2 0 0 0 3 1 1 1 0 0 2 2 1 1 0 0 2 1 1 1 1 0 1 1 1 1 1 1
Translation into explicit definition:
APx =: 3 : 0 NB. (n) APx (p)
:
all =. i.0 0
while. y
do. if. x>y do. parts =. 1 + (x-y) APx y
else. parts =. (x=y) # ,: y#1
end.
y =. y-1
all =. parts , all
end.
)
This uses 20% less space than AP but is much slower and longer. So, it will bow out.
Tacit AP can speed up greatly by exiting when either input is 0 or 1:
AP =: ;@All ELSE N WHEN Under2
Under2 =: 2 > <.
N =: ,:@{.~
This revised version will be used for comparisons and is listed completely in the Appendix.
It runs 3 times faster than Skiena but needs about 30% more space at n=65. Yet, AP can do n = 70, whereas Skiena runs out of memory. It is also faster than AP1 and AP2 at n = 70 in less space, although much slower at n=65. Which is better? See Comparisons.
APr
Here is a simple recursive definition for two inputs: n and a lead part. Explicitly:
APrx =: 3 : 0 NB. (n) APrx (lead)
:
all =. i.0,x
while. y>1
do. all =. all , y ,. n APrx y <. n=.x-y
y =. y-1
end.
all,1
)
Note that it skips the loop whenever y is 1 and just appends the last row of 1s.
Example: 6 APrx 6 or APrx~6 is the same as Knuth 6.
Albeit short, this program is much slower than AP yet much slimmer in space until n = 70 when it exceeds memory. So, it will drop out.
A tacit ambivalent version is 40% faster in the same space and can do n = 70:
APr =: ;@All ELSE Ones WHEN Under2 NB. (n) APr (lead) or APr (n)
Ones =: ,:@#
Under2 =: 2 > ]
All =: Leads@] Parts EACH Ns
Parts =: [ ,. ] APr <.
Leads =: - i.
Ns =: - Leads
This can be improved further by generating 1s separately:
APr =: <@Ones ;@, Allbut1s NB. (n) APr (lead) or APr (n) Ones =: ,:@# 1: Allbut1s =: Parts EACH Leads Parts =: ] ,. - APr - <. ] Leads =: 2 }. i. , ]
It is now competitive with AP in speed, but 25% fatter, and shorter. Indeed, this is the shortest program among those here that can perform high n.
APm
Another simple recursive definition uses previous partitions as a “memo” to produce the next:
APm =: 3 : 0 NB. APm (n)
all =. i.0,n=.y
while. n>1
do. memo =: APm y-n
leads =. {."1 memo
drop =. leads i. n <. y-n
all =. all , n ,. drop }. memo
n =. n-1
end.
all,1
)
Note that global memo cuts down space significantly. Still, it’s way too slow.
J’s built-in Memo adverb M. does better. See APdb later in Other Programs.
APh
This approach produces all partitions in two halves, recursively, in an ambivalent program. For the bottom half, it recurses when n > lead part; and for the top half, it recurses with n as both inputs.
APh =: [ ;@Parts Ns NB. APh (n) or (n) APh (lead)
Ns =: -@<. {. i.@[
Parts =: ] <@Part"0 -
Part =: ] ,. Recurse ELSE Empty WHEN Zero ELSE Ones WHEN One
Recurse =: [ APh [ ELSE ] WHEN >
Empty =: ,:@i.@0:
Zero =: = 0:
Ones =: ,:@#
One =: 1 = ]
Examples of its subprogram for two inputs:
0 Part 6
6
1 Part 5
5 1
2 Part 4
4 2 0
4 1 1
3 Part 3
3 3 0 0
3 2 1 0
3 1 1 1
4 Part 2
2 2 2 0 0
2 2 1 1 0
2 1 1 1 1
5 Part 1
1 1 1 1 1 1
APh 6 assembles these results into one table of descending partitions (same as APr 6).
Include an exit for 2 and build an array with leading 2s directly:
Part=: ] ,. Recurse ELSE Empty WHEN Zero ELSE Twos WHEN Two ELSE Ones WHEN One
Two =: 2 = ]
Twos =: Build ^: Inputs
Build =: 1 ,~ 2 ,. ] ,. 0:
Inputs =: <.@% ` Start
Start =: ,:@:>:@i.@|~
This program is now much faster in equal space but quite longer.
APn
Another approach constructs a table of partitions horizontally by nesting columns recursively:
APnx =: 3 : 0 NB. APnx (n)
is =. i.y
; is Nest EACH y-is
)
Nest =: 3 : 0
:
if. y=1 do. 1,x#1 return. end.
if. x=0 do. ,:,y
else. min =. – x <. y
is =. min {. i.x
y ,. ; is Nest EACH x-is
end.
)
A tacit version is shorter and much faster:
APn =: [ ;@Parts <. NB. APn (n) Parts =: Is Nest EACH Ns@] Is =: i.@] + - Ns =: - i. Nest =: Join ELSE N WHEN Zero ELSE Ones WHEN One Join =: ] ,. APn N =: ,:@,@] Zero =: =0: Ones =: 1,# One =: 1=]
APapr
Finally, consider this: immediately halve input n to become the largest second part, subtract it from n to get the first part, iterate through decrements of first and second parts respectively, and call APr (presented previously) to attach a sub-array of partitions for the remaining newn with the leading part as the smaller of second and newn:
APapr =: 3 : 0 NB. APapr (n) y =. <.-:n=.y all =. Ns1s n-i.n for_second. 2 To y do. for_first. |.second To n-second do. all =. all,first,.second,.newn APr second<.newn=.n-first+second end. end. )
For efficiency, all partitions with a second part 1 and 0 are done at once separately in a table:
Ns1s =: ,. (</ }:) NB. Ns1s (n-i.n) To =: }. i. , ]
Example:
APapr 6 6 0 0 0 0 0 5 1 0 0 0 0 4 1 1 0 0 0 3 1 1 1 0 0 2 1 1 1 1 0 1 1 1 1 1 1 4 2 0 0 0 0 3 2 1 0 0 0 2 2 1 1 0 0 2 2 2 0 0 0 3 3 0 0 0 0
Note the unorthodox order: by increasing second part. This program is faster than Kelleher, using a little more space, and much shorter (even counting APr). Despite relying on a sub-program that itself can compute all partitions, it competes very well at high n.
Comparisons
Now compare the most competitive programs here for n = 65 by ratios of time, space, and length – where 1.00 is best. Finally, sum the ratios for a simple composite of overall program efficiency:
Time Space Length Overall Skiena 12.37 1.46 1.00 14.83 Knuth 5.06 1.00 7.94 14.00 Boss 1.52 3.00 3.00 7.52 Hui 23.89 5.41 2.38 31.69 AP1 1.00 2.34 2.79 6.13 AP2 1.00 2.34 2.56 5.90 Kelleher 4.41 1.00 6.35 11.77 AP 3.57 1.89 1.59 7.05 APr 3.47 2.34 1.15 6.95 APh 3.64 2.34 3.44 9.42 APn 4.76 2.34 1.85 8.96 APapr 3.37 1.03 4.12 8.51
So, AP2 is the winner, and AP1 second.
For n = 70 , Skiena, Boss and Hui cannot participate on my computer because they run out of memory. Timing results for most other programs have large variance since they are pushing space limits. Within this uncertain repeatability, APapr emerges as the fastest and AP becomes the new winner overall. In order, the top six are: AP, APr, APapr, Kelleherx, AP2, AP1.
Considering only time and space, the top six are: APapr, Kelleherx, Knuth, AP, APr, AP1.
No program above can do n = 75 on my laptop. Timing ratios on other computers differ somewhat but indicate that AP1 is the fastest, with AP2 close behind.
Readers may be able to run higher n, may want to weight the three criteria differently, and may add other criteria, such as a measure of clarity.
See Appendix for benchmark details and copies of the best programs for convenient use.
Number of Partitions
For any program above, the number of partitions can be counted perfunctorily from the resulting table of partitions. For example: #AP 6 is 11. A list of partition numbers:
#@AP"0 i.20 1 1 2 3 5 7 11 15 22 30 42 56 77 101 135 176 231 297 385 490
There is an issue about how to count the number of partitions for n = 0:
$Skiena~ 0 1 0 NB. #Knuth 0 does not compute $Boss 0 0 $Hui 0 1 0 $AP1 0 1 $AP2 0 0 $Kelleher 0 1 1 $AP 0 1 0 $APr 0 1 0 $APapr 0 0 1
Since the sum of an empty list is 0, one can argue that there are zero partitions of 0. One can also argue that there is one partition of zero: the empty partition. See oeis.org/wiki/Partition function.
So, what should it be – an empty list or a 1 by 0 empty table or just 0?
Of course, the pertinent problem is how to compute number of partitions efficiently for large n without the effort of generating them.
P
To begin with, here is a clean program to compute number of partitions of integer n with k parts, albeit inefficiently due to recursion:
P =: Recur ELSE = WHEN Done NB. (n) P (k)
Recur =: P&<: + - P ]
Done =: <: +. 0 = ]
An alternative definition:
P =: Recur ELSE ] WHEN (2>]) ELSE = WHEN <: NB. (n) P (k)
For example, number of partitions of 10 with 4 parts:
10 P 4 9
The total number of partitions of n with less than k parts is +/ n P"0 i.>:k.
A table of P for both n and k from 0 to 10:
P"0/~ i.11 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 2 1 1 0 0 0 0 0 0 0 1 2 2 1 1 0 0 0 0 0 0 1 3 3 2 1 1 0 0 0 0 0 1 3 4 3 2 1 1 0 0 0 0 1 4 5 5 3 2 1 1 0 0 0 1 4 7 6 5 3 2 1 1 0 0 1 5 8 9 7 5 3 2 1 1
Notice that the nth row sum is the number of all partitions of n.
Pnk
An efficient program to build this table iteratively by adding the shifted reverse diagonal to the last row:
TP =: One , 0 ,. Row ^: (Repeat`One) NB. TP (n)
Repeat =: 0 >. <:
One =: ,:@,@1:
Row =: , Last + Diag , 0:
Last =: 0 , {:
Diag =: (<0 1) |: |.
Index the table to get number of partitions of n with k parts:
Pnk =: <@, { TP@[ NB. (n) Pnk (k) or Pnk (n)
Example:
10 Pnk 4 9
Since Pnk is ambivalent, Pnk(n) is the last row, and its sum is the number of all partitions:
NP =: +/@Pnk NB. NP (n)
Example:
NP 10 42
First 20 numbers of all partitions:
NP"0 i.20 1 1 2 3 5 7 11 15 22 30 42 56 77 101 135 176 231 297 385 490
Larger numbers:
n ,: NP"0 n =. 5*i.15 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 1 7 42 176 627 1958 5604 14883 37338 89134 204226 451276 966467 2012558 4087968
Growth factors:
Growth =: }. % }:
5j2 ": Growth NP"0 i.15
1.00 2.00 1.50 1.67 1.40 1.57 1.36 1.47 1.36 1.40 1.33 1.38 1.31 1.34
load 'plot'
plot Growth NP"0 i.50
Growth NP"0 n=.1000+i.10 1.04035 1.04033 1.04031 1.04029 1.04027 1.04025 1.04023 1.04021 1.04019
For very large n, use Hui’s efficient pnv program in [6] based on Euler’s recurrence relation to see convergence to an asymptote approaching 1.
Other Programs
Several other programs that generate all integer partitions have not been mentioned yet because they are not nearly competitive with those above. With respect for their role in research and development, they are summarized as follows:
Reingold
Reingold et al. [10] described an algorithm for generating tuples with ascending parts. Coded explicitly in J:
Reingold =: 3 : 0 NB. Reingold (n)
n =. y
m =. 1
ps =. i.0 0
while. n>:m
do. ps =. ps , n Parts m
m =. m + 1
end.
ps
)
Parts =: 3 : 0 NB. (n) Parts (m)
:
n =. x
m =. y
ps =. ,: 1 + (-m) {. n-m
while. +./1<p-~{:p=.{:ps
do. ps =. ps , Next p
end.
)
Next =: 3 : 0 NB. Next (p)
p =. y
n =. +/p
m =. #p
i =. <: 0 i.~ 2 <: p-~{:p
p =. (1+i{p) (i }. i.m)} p
p =. (n-+/}:p) (m-1)} p
)
For example:
Reingold 6 6 0 0 0 0 0 1 5 0 0 0 0 2 4 0 0 0 0 3 3 0 0 0 0 1 1 4 0 0 0 1 2 3 0 0 0 2 2 2 0 0 0 1 1 1 3 0 0 1 1 2 2 0 0 1 1 1 1 2 0 1 1 1 1 1 1
The program is lengthy, four times slower than Knuth at n = 65 in twice the space, and runs for hours at n = 70. Tacit translations (for both ascending and descending parts) use 30% less space but are ten times slower.
Reiter
Reiter [11] presented a recursive program in APL that uses three inputs (in one list): integer n, number of parts p and smallest part s. It “splits the computation depending on whether there are partitions into one, two, or more parts. The case of two parts is not necessary to the correctness of the algorithm, but it does improve the efficiency of the algorithm. When the number of parts is three or more, then a loop catenates blocks of partitions of smaller (or equal) numbers into a smaller number of parts with the smallest number allowed being the index of the loop.” [11 page 8] In J:
Reiter =: ;@Blocks ELSE (,@N) WHEN (P < 2:) NB. Reiter (n,p,s)
Blocks =: <@(S,.Reiter)"1 @ NPSs
NPSs =: (N-Ss) ,. <:@P ,. Ss
N =: {.
P =: {.@}.
S =: {:
Ss =: S To N Q P
Q =: <.@%
To =: [ + i.@>:@-~
Example of partitions of 10 with 4 parts, starting with 1:
Reiter 10 4 1 1 1 1 7 1 1 2 6 1 1 3 5 1 1 4 4 1 2 2 5 1 2 3 4 1 3 3 3 2 2 2 4 2 2 3 3
All partitions of n are produced by inputs (n,n,0). For example: Reiter 6 6 0 .
This program runs slower than Skiena but needs less space. Unfortunately, it is much longer.
Peelle
Here is my doubly recursive program based on P (a few pages earlier) that uses two fundamental functions to generate all partitions – appending a 1 and adding 1:
Peelle =: ;@All NB. Peelle (n) All =: Parts EACH >:@i. Parts =: Recurse ELSE Empty WHEN < ELSE N WHEN One ELSE Ones WHEN = Recurse =: Append1 , Plus1 Append1 =: Parts&<: ,. 1: Plus1 =: 1 + - Parts ] Empty =: 0 i.@, ] N =: ,:@,@[ One =: 1 = ] Ones =: # 1:
This program is twice as fast as Skiena, in the same space as AP, a pinch shorter than Reiter and quite competitive at n=70 – notably faster, slimmer, and shorter than AP1 and AP2.
Further, here are several unusual (and inefficient) approaches that may be of interest to the curious:
APod
An odometer approach entails selecting unique lists of partitions with sorted parts from a table of consecutive integers represented as lists in base n + 1.
The program is loopless and short, but indulgent:
APod =: ~.@Select (Odometer >:)
Odometer =: # #: i.@^~ NB. (n) Odometer (n+1)
Select =: Sort"1@Parts
Parts =: IsSum"1 # ]
Sort =: /:~
IsSum =: = +/
Lexical order, as on an odometer:
APod 6 0 0 0 0 0 6 0 0 0 0 1 5 0 0 0 0 2 4 0 0 0 0 3 3 0 0 0 1 1 4 0 0 0 1 2 3 0 0 0 2 2 2 0 0 1 1 1 3 0 0 1 1 2 2 0 1 1 1 1 2 1 1 1 1 1 1
Revise it to be 2.5 times faster:
APod =: N , (~.@Select Odometer~)
N =: - {. ]
Using multiple bases is 64 times faster still:
APod =: ~.@Select Odometer
Odometer =: Encode@Bases NB. Odometer (n)
Encode =: #: i.@(*/)
Bases =: 1 + Diagonal@i.
Diagonal =: {"0 1 Copies@:>:
Copies =: |. #"0 1 ]
6 times faster again (with left-justified result):
APod =: ;@(<@Parts"0 Ns)
Parts =: 1 + - ~.@Select Odometer NB. (n) Parts (p)
Odometer =: Encode@Bases NB. (n) Odometer (p)
Encode =: #: i.@(*/)
Bases =: (-i.) Q |.@Ns@]
Q =: <.@%
Even though this is hundreds times faster than the initial definition, it is so indulgent that it runs out of memory at about n = 30.
AP9s
Another approach encodes only multiples of 9 (instead of a full odometer) into base-10 digits, selects ascending lists, then selects lists that sum to n. How crude!
AP9s =: All ELSE i. WHEN (=0:)
All =: Parts Ascend@Encode10
Encode10 =: #&10 #: (+ 9 Multiples <:)
Multiples =: * i.@(10&^)
Ascend =: #~ (-: Sort)"1
Sort =: /:~
Parts =: IsSum"1 # ]
IsSum =: = +/
This program is very slow and space-consuming – only able to do up to n = 8.
APid
This next approach iterates adding an identity matrix to generate all possible new partitions: Start with an empty 1 by 0 array. Iteratively join a left column of 0s then a top row of 1s to ascending (sorted) lists in tables created by adding each row of an appropriate size identity matrix to each row in the previous array. Finally, remove duplicate partitions from the result.
APid =: ~.@All^:(]`Empty)
Empty =: ,:@i.@0:
All =: 1 , 0 ,. Parts
Parts =: Sort"1@(,/)@:(Plus"1)
Sort =: /:~
Plus =: +"1 Id
Id =: =@i.@#
APid 6
1 1 1 1 1 1 NB. Reverse lexical order
0 1 1 1 1 2
0 0 1 1 2 2
0 0 1 1 1 3
0 0 0 2 2 2
0 0 0 1 2 3
0 0 0 1 1 4
0 0 0 0 3 3
0 0 0 0 2 4
0 0 0 0 1 5
0 0 0 0 0 6
This program runs out of memory at n = 45.
Alternative: Use boxes and trim the identity matrix to add 1s only to the last unique part, so there is no need to sort.
APid =: ~.@All^:(]`Empty)
Empty =: ,:@i.@0:
All =: 1 , 0 ,. ;@:Parts
Parts =: <@Plus"1
Plus =: +"1 Id
Id =: (i: ~. -. 0:) =/ i.@#
This is almost 10 times faster and uses less than 10% space, but runs out of memory at n = 60.
APpnk
Here is an approach that computes the number of partitions in advance for a given n and k in order to generate partitions iteratively in k-tuples. It uses program Pnk (from Number of Partitions earlier) to determine how many times to iterate.
APpnk =: ;@All NB. APpnk (n)
All =: Parts EACH Ns
Ns =: >:@i.
Parts =: Next^:(Pnks`Start) NB. (n) Parts (k)
Pnks =: i.@Pnk NB. (n) Pnk (k)
Start =: 1 + ] {. -
Next =: Front To Body ]
To =: Tos i. 1:
Tos =: 1 < (- ~{.)
Body =: Copies , Back
Back =: >:@[ }. ]
Copies =: [ # >:@{
Front =: (- +/) , ]
(APpnk -: AP) 6
1
This program is shorter and faster than Knuth and Kelleher at n=65, but requires more space. It is even more competitive at n=70, although it seems like cheating.
APdb
Another approach updates a database every time the program is used anew. A global variable contains previously executed results for smaller n and k that can be looked up directly instead of re-computed.
First, initialize a global boxed matrix:
parts =: 1 1 $ <i.1 0
Define the program to immediately extend the matrix with empty boxes if either input n or k is larger than its shape. Then look up the nth row and kth column. If it is empty, compute the result and update the global matrix; otherwise, open it as the result.
APdb =: 3 : 0 NB. (n) APdb (k) or APdb~ (n)
:
nk =. x,y
if. +./ nk >: $parts do. parts =: (nk+1) {. parts end. NB. extend
p =. (<nk) { parts NB. look up
if. p=a: do. p =. < x APrdb y NB. compute
parts =: p (<nk)} parts NB. update
end.
>p
)
The following co-program used above is a modification of APr that computes all partitions of n in k parts recursively but calls APdb (instead of itself) to look up a sub-result if it is already known.
APrdb =: ;@All ELSE Ones WHEN Under2 NB. (n) APrdb (lead)
Ones =: ,:@#
Under2 =: 2 > ]
All =: Leads@] <@Parts"0 Ns
Parts =: [ ,. ] APdb <. NB. APdb instead of APr
Leads =: - i.
Ns =: - Leads
Using a database is advantageous whenever partitions below n and k must be re-computed quickly. APdb is also quite speedy for new n and k up to about n = 60. For instance, APdb~60 is about 4 times faster than APr 60. However, by n = 65, it becomes sluggish and demands about 50% more space even though only about 1r6 of the database is filled up.
It is easier to use J’s built-in Memo adverb M. to gain about the same advantage.
For example: NB. APr =: ;@All M. ELSE Ones WHEN Under2
Then APr 60 would be 4 times faster, but still more than 4 times slower for n = 65.
Other Representations
So far, a natural representation has been used for a partition – that is, a list of positive integers (without interspersed + symbols). For example, a partition of 15: 5 3 3 1 1 1 1 or 1 1 1 1 3 3 5.
Other possible representations include: base (or standard), frequency (and multiplicity), and Ferrers dot diagram.
Base
The numerals of a partition can be compressed into digits of a single number in base 10, as in 5331111 or 1111335. This might be advantageous for certain algorithms, such as selecting multiples of 9 (see AP9s above). However, if any part exceeds 9, this representation becomes awkward, necessitating pairs of digits, or triples for n > 99, etc.
To convert from natural to base representation, use J’s Decode with base 10 for partitions such as:
10 #. 5 3 3 1 1 1 1 10 #. 1 1 1 1 3 3 5 5331111 1111335
To convert from base to natural, use its inverse Encode with the appropriate base for each digit:
(7#10) #: 5331111 5 3 3 1 1 1 1
Or use J’s Power to find the inverse:
10&#. ^:_1 ] 5331111 5 3 3 1 1 1 1
Frequency
A partition can be represented as a list of frequencies of parts from 1 to the largest.
To convert from natural to frequency representation, use this program:
Freq =: +/@(=/) >:@i.@(>./)
Example:
Freq 5 3 3 1 1 1 1 NB. Freq 1 1 1 1 3 3 5 4 0 2 0 1
To convert from frequency to natural:
FreqInv =: # >:@i.@#
FreqInv 4 0 2 0 1
1 1 1 1 3 3 5
Or include its inverse in the definition:
Freqs =: Freq :. FreqInv
Freqs^:_1 ] 4 0 2 0 1
1 1 1 1 3 3 5
For any partition of n (in any order), n equals the sum of frequencies times integers 1 to the largest part, respectively:
p =. >:?10#10
n =. +/p
fs =. Freq p NB. fs =. Freqs p
n = +/fs*>:i.>./p
1
Multiplicity
Alternatively, a partition can be represented simply by multiples of its unique parts.
For example, 4 2 1 multiples of unique parts 1 3 5 represents 1 1 1 1 3 3 5.
Convert from natural to multiplicity:
+/"1 =1 1 1 1 3 3 5 4 2 1
Convert from multiplicity to natural representation:
1 3 5 #~ 4 2 1 1 1 1 1 3 3 5
Define a program with its inverse:
Multi =: +/"1@= :. (#~)
Both conversions:
Multi 1 1 1 1 3 3 5
4 2 1
1 3 5 Multi^:_1 ] 4 2 1
1 1 1 1 3 3 5
Any sorted partition p is identical to the inverse conversion of its conversion:
p -: (~. Multi^:_1 Multi) p =. /:~ >:?10#10 1
Ferrers Diagram
Another representation devised by Norman M. Ferrers provides visualization of a partition via rows of dots (or any symbol) for each part. See [2] or [12]. For example, 5 3 3 1 1 1 1 would be:
* * * * *
* * *
* * *
*
*
*
*
A program to produce Boolean indices for such a diagram:
Ferrers =: (# 1:)"0 NB. Ferrers =: >/ i.@(>./)
Example:
Ferrers 5 3 3 1 1 1 1 ' *' {~ Ferrers 5 3 3 1 1 1 1
1 1 1 1 1 *****
1 1 1 0 0 ***
1 1 1 0 0 ***
1 0 0 0 0 *
1 0 0 0 0 *
1 0 0 0 0 *
1 0 0 0 0 *
Include its inverse:
Ferrers =: (# 1:)"0 :. (+/"1)
To produce a conjugate partition, simply sum the rows (vertically):
+/ Ferrers 5 3 3 1 1 1 1 7 3 3 1 1
Fractal Patterns
Due to the recursive structure of partitions, it is not surprising to find fractal-like patterns. See [13].
load 'graph'
viewmat 1 = Knuth 10 NB. white 1s
viewmat 1 = AP 20
Similarly, view other parts, such as viewmat 2 = AP 20 or all parts in colors: viewmat AP 10
Summary
J programs for generating all integer partitions were presented and compared for the first time.
Most programs and some algorithms are new. The best program was determined (within my computing constraints) for equally-weighted criteria that included program length as well as speed and space. Translations between tacit and explicit definitions were often supplied and contrasted without due explication. To follow up, see References.
Acknowledgements
Thanks to Professor Emeritus Murray Eisenberg (UMass Mathematics & Statistics Department) for his helpful critique and for obtaining benchmarks on his iMac with J7.01. Thanks also to Boyko Bantchev for valuable feedback and generous contributions, to Devon McCormick (Daedalus Systems, New York) for checking performance of programs, and to Professor Cliff Reiter (LaFayette College Mathematics Department) for continuing attention.
References
[0] www.Jsoftware.com
[1] S. Skiena, Implementing Discrete Mathematics … with Mathematica, Addison-Wesley, 1990
[2] D. E. Knuth, The Art of Computer Programming Vol.4A 7.2.1.4, Addison-Wesley, 2005-2011
[3] C. F. Hindenburg, Infinitomii Dignitatum Exponentis Indeterminati (Göttingen) pp. 73-91, 1779
[4] R. Hui, www.jsoftware.com/pipermail/general/2005-June/023191.html
[5] R.E.Boss, "Partitions of numbers: An efficient algorithm in J", __Vector__ 23.4 pp. 121-131, 2008
[6] R. Hui, Jsoftware/Jwiki/Essays/Partitions, 2008-2011
[7] H. A. Peelle, Jsoftware/Jwiki/Essays/AllPartitions, 2011
[8] J. Kelleher & B. O’Sullivan, "Generating All Partitions: A Comparison of Two Encodings",
arxiv.org > cs > arxiv: 0909.2331 [cs.D5], 2009
[9] A. Zoghbi & I. Stojmenovic, “Fast Algorithms for Generating Integer Partitions”,
__Int. J. Computer Math.__, Vol. 70 pp. 319-332, 1998
[10] E. Reingold, J. Nievergelt, & N. Deo, Combinatorial Algorithms, Prentice-Hall, 1977
[11] C. Reiter, “Random Markov Matrices and Partitions of Integers”, __APL Quote-Quad__, Vol. 22, No. 3, pp. 7-8, March 1992
[12] E. W. Weisstein, Mathworld.wolfram.com/PartitionFunctionP
[13] A. Salerno, "Partition Numbers Unveiled as Fractal", __MAA Focus__, April-May 2011
maa.org/focus.html
Appendix
Benchmarks for time and space were obtained on a Dell Latitude E6410 laptop (64-bit OS M620 at 2.67 GHz with 4GB RAM) running J6.02 under Windows 7.
Spread (Length) = total number of characters in a program definition body, ignoring spaces, and counting only 1 for each name.
For a table t of benchmarks, the table of ratios (in Comparisons) is: (,. +/"1) t %"1 <./t
Utility programs:
Time =: 6 !: 2
Space =: 7 !: 2
Test =: (-: ~.) *. +/"1 *./ . = {:@$ NB. all unique and all rows sum = n
Sort =: /:~"1
Example tests:
*./Test@Knuth"0 }.i.66
1
*./Test@Boss"0 i.66
1
*./Test@Hui"0 i.66
1
*./Test@AP"0 i.66
1
Example plots of Time and Space:
load 'plot'
plot n ; Time"1 'AP2 ' ,"1 ":,.n=.5*i.14
plot n ; Space"1 'AP2 ' ,"1 ":,.n=.5*i.14
Program Definitions
Definitions of the best programs in Comparisons are shown below for convenient copying and pasting into separate scripts:
AP1 =: ;@All ` Ns @. (< 2:) NB. AP1 (n)
All =: Ns ,.&.> Parts/@Mins
Ns =: >:@i.
Mins =: Smaller@}.@i. , 0:
Smaller =: <. |.
Parts =: Next ` Start @. Zero
Zero =: 0 e. ]
Start =: 1 ; ,@0:
Next =: ;@New ; ]
New =: Ns@[ Join&.> {.
Join =: [ ,. Select # ]
Select =: >: {."1
---
AP2 =: ;@All NB. AP2 (n)
All =: Ns ,.&.> Parts
Ns =: >:@i.
Parts =: Build^:N Start
Start =: <@,:@i.@0:
N =: 0 >. <:@[
Build =: <@;@Next , ]
Next =: Lead Ps ]
Lead =: Min #
Min =: - <. ]
Ps =: Ns@[ Join&.> {.
Join =: [ ,. Select # ]
Select =: >: {."1
---
AP =: ;@All ` N @. Under2 NB. AP (n) or (n) AP (p)
Under2 =: 2 > <.
N =: ,:@{.~
All =: Parts &.> >:@i.
Parts =: Ones ` Plus1 @. >
Ones =: = # ] ,:@# 1:
Plus1 =: 1 + - AP ]
---
APr =: <@Ones ;@, Allbut1s NB. APr (n) or (n) APr (lead) Ones =: ,:@# 1: Allbut1s =: Parts &.> Leads Parts =: ] ,. - APr - <. ] Leads =: 2 }. i. , ]
---
Kelleherx =: 3 : 0 NB. Kelleherx (n)
a =. 0,y#0
k =. 1
y =. y-1
all =. i.0 0
while. k do.
k =. k-1
x =. 1+k{a
while. y>:2*x do.
a =. x k}a
y =. y-x
k =. k+1
end.
l =. k+1
while. x<:y do.
a =. x k}a
a =. y l}a
all =. all,a{.~k+2
x =. x+1
y =. y-1
end.
a =. a k}~x+y
y =. x+y-1
all =. all,a{.~k+1
end.
)
---
APapr =: 3 : 0 NB. APapr (n) y =. <.-:n=.y all =. Ns1s n-i.n for_second. 2 To y do. for_first. |. second To n-second do. all =. all,first,.second,.newn APr second<.newn=.n-first+second end. end. ) N1s =: ,. (</ }:) To =: }. i.@>: NB. uses APr (above)