Perhaps here is a better way to describe the problem I was trying to solve:
1. I have y unique objects. I have x containers. I want to find all the different ways I can put those y unique objects into x containers. 2. A single trial is defined as distributing *all* y of the objects into all of the x containers. After the trial, the objects are removed from the containers and redistributed in the containers for the next trial. Objects can not be duplicated. 3. Empty containers are not allowed in any trial. There must be at least one object in every container in every trial. 4. There is no order requirement for objects in a container. If two trials are identical except for the order of the objects in any of the containers, those two trials are considered as one trial. 5. There is no order requirement for the containers. If two trials are identical in that both trials have the same number of objects in each of the x containers, but the order of the containers is different, those two trials are considered as one trial. Skip Skip Cave Cave Consulting LLC On Fri, Oct 20, 2017 at 3:54 AM, Raul Miller <[email protected]> wrote: > With that description, I think I might do something like this: > > nparts=: ~.@([: /:~"1 i.@] </."1~ [ #.^:_1 [ i.@^ ]) > 2 nparts 3 > ┌───┬─────┐ > │ │0 1 2│ > ├───┼─────┤ > │0 1│2 │ > ├───┼─────┤ > │0 2│1 │ > ├───┼─────┤ > │0 │1 2 │ > └───┴─────┘ > > I should be able to do something more efficient, but before I attempt > that, I would like to clarify something: > > In your examples, you do not have any empty partitions, so is that a > part of the specification also? > > I am unsure if I should be paying close attention to your examples > because you said "The order of the objects in each partition is not > important" but your examples also omit partitions which contain the > objects out of order. > > Actually... there's several kinds of order here which we could be > discussing: > > (1) the order of objects within a partition. > (2) the order of objects across partitions. > (3) the order of partitions. > > In other words: > > NB. (1) the order of objects within a partition > \:~each 2 nparts 3 > ┌───┬─────┐ > │ │2 1 0│ > ├───┼─────┤ > │1 0│2 │ > ├───┼─────┤ > │2 0│1 │ > ├───┼─────┤ > │0 │2 1 │ > └───┴─────┘ > > NB. (2) the order of objects across partitions > 2 ~.@([: \:~"1 i.@] </."1~ [ #.^:_1 [ i.@^ ]) 3 > ┌─────┬───┐ > │0 1 2│ │ > ├─────┼───┤ > │2 │0 1│ > ├─────┼───┤ > │1 │0 2│ > ├─────┼───┤ > │1 2 │0 │ > └─────┴───┘ > > NB. (3) the order of partitions > 2 ((i.@[ ,"1 [ #.^:_1 i.@^) <@}./."1 {. , i.@]) 3 > ┌─────┬─────┐ > │0 1 2│ │ > ├─────┼─────┤ > │0 1 │2 │ > ├─────┼─────┤ > │0 2 │1 │ > ├─────┼─────┤ > │0 │1 2 │ > ├─────┼─────┤ > │1 2 │0 │ > ├─────┼─────┤ > │1 │0 2 │ > ├─────┼─────┤ > │2 │0 1 │ > ├─────┼─────┤ > │ │0 1 2│ > └─────┴─────┘ > > I have presumed that you are thinking of both the partition contents > and the partitions themselves as sets. In other words, I think that > none of these orders matter. But... this kind of thing is worth > verifying? > > Thanks, > > -- > Raul > > On Fri, Oct 20, 2017 at 12:19 AM, 'Skip Cave' via Programming > <[email protected]> wrote: > > Mike, > > > > I wasn't very thorough in my definition of the original problem. I > thought > > that the example I gave was enough to clarify the requirements, but > looking > > back, more definition would have been good. > > > > The original problem I posted was to develop a dyadic verb that would > take > > y objects and show all the ways that those y objects could be partitioned > > into x groups. Each partition set must include all objects exactly once. > > Duplication of objects is not allowed. The order of the objects in each > > partition is not important. > > > > Erling got the right idea in his previous post: > > > > par=: 4 : '(1,.2</\"1(i.x)#/~(y=+/"1 o)#o=.((x$v)#:i.v^x){1+i.v=.1+ > > y-x)<;.1[1+i.y' > > > > 2 par 3 > > ┌───┬───┐ > > │1 │2 3│ > > ├───┼───┤ > > │1 2│3 │ > > └───┴───┘ > > 2 par 4 > > ┌─────┬─────┐ > > │1 │2 3 4│ > > ├─────┼─────┤ > > │1 2 │3 4 │ > > ├─────┼─────┤ > > │1 2 3│4 │ > > └─────┴─────┘ > > 2 par 5 > > ┌───────┬───────┐ > > │1 │2 3 4 5│ > > ├───────┼───────┤ > > │1 2 │3 4 5 │ > > ├───────┼───────┤ > > │1 2 3 │4 5 │ > > ├───────┼───────┤ > > │1 2 3 4│5 │ > > └───────┴───────┘ > > 3 par 4 > > ┌───┬───┬───┐ > > │1 │2 │3 4│ > > ├───┼───┼───┤ > > │1 │2 3│4 │ > > ├───┼───┼───┤ > > │1 2│3 │4 │ > > └───┴───┴───┘ > > 3 par 5 > > ┌─────┬─────┬─────┐ > > │1 │2 │3 4 5│ > > ├─────┼─────┼─────┤ > > │1 │2 3 │4 5 │ > > ├─────┼─────┼─────┤ > > │1 │2 3 4│5 │ > > ├─────┼─────┼─────┤ > > │1 2 │3 │4 5 │ > > ├─────┼─────┼─────┤ > > │1 2 │3 4 │5 │ > > ├─────┼─────┼─────┤ > > │1 2 3│4 │5 │ > > └─────┴─────┴─────┘ > > 3 par 6 > > ┌───────┬───────┬───────┐ > > │1 │2 │3 4 5 6│ > > ├───────┼───────┼───────┤ > > │1 │2 3 │4 5 6 │ > > ├───────┼───────┼───────┤ > > │1 │2 3 4 │5 6 │ > > ├───────┼───────┼───────┤ > > │1 │2 3 4 5│6 │ > > ├───────┼───────┼───────┤ > > │1 2 │3 │4 5 6 │ > > ├───────┼───────┼───────┤ > > │1 2 │3 4 │5 6 │ > > ├───────┼───────┼───────┤ > > │1 2 │3 4 5 │6 │ > > ├───────┼───────┼───────┤ > > │1 2 3 │4 │5 6 │ > > ├───────┼───────┼───────┤ > > │1 2 3 │4 5 │6 │ > > ├───────┼───────┼───────┤ > > │1 2 3 4│5 │6 │ > > └───────┴───────┴───────┘ > > > > Skip Cave > > Cave Consulting LLC > > > > On Thu, Oct 19, 2017 at 5:47 PM, 'Mike Day' via Programming < > > [email protected]> wrote: > > > >> Skip, in your actual Quora Problem, why not include other triads, > >> such as 1 1 24, 2 3 4 etc; or, otherwise, why include both 2 4 3 > and 4 > >> 2 3 ? > >> > >> Anyway, this is (quite) short and brutish but not too nasty to solve > your > >> Quora problem for quite small numbers and numbers of factors: > >> > >> |: 24 ([ ( (= */"1)#]) [:>:[#.inv i.@^ ) 3 NB. transpose > gratuitous! > >> 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 4 4 6 6 6 8 8 12 12 24 > >> 1 2 3 4 6 8 12 24 1 2 3 4 6 12 1 2 4 8 1 2 3 6 1 2 4 1 3 1 2 1 > >> 24 12 8 6 4 3 2 1 12 6 4 3 2 1 8 4 2 1 6 3 2 1 4 2 1 3 1 2 1 1 > >> > >> It builds triads 1 1 1 , 1 1 2 ,...1 1 24, ... up to 24 24 24, and keeps > >> > >> just those whose product is 24. > >> > >> > >> No points for space or time, filtering 30 out of 13824 candidates, but > it's > >> > >> quite straightforward, and it does yield all 30 permutations, which > some > >> > >> of the Quora corresondents appear to consider the requirement. > >> > >> > >> NB - it's not clear to me what the problem actually is - is 30 the > required > >> > >> answer (number of permutations of 3 suitable factors), or 6 (number of > >> > >> combinations of same)? > >> > >> > >> Mike > >> > >> > >> > >> > >>> > >>> > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
