Today I discovered from my archives that I'd revisited my old 1970 work in 2018 and got much further with it than I recall. The code I wrote back then has verbs which convert between (A.) and my 1970 treatment, as well as dissecting my algorithm in an explainable way, which doesn't invoke too much fancy group theory.
It'll take me a few days to decipher what I did back then and compare it with the Wikipedia articles and Norman Thomson's Perming and Combing article in "Fifty Shades". I want to convince myself that I've simply re-invented the Lehmer wheel, in which case I'll cheerfully abandon my approach in favor of published descriptions based on Lehmer codes. I'll also consider if your (A.&i.&.|.~ !) sheds light on the problem of an efficient J implementation. The pattern it generates doesn't ring a bell with me, I fear. On Sun, 20 Mar 2022 at 23:19, Raul Miller <[email protected]> wrote: > After thinking about this, I am wondering whether this would match > your concept of Lehmer codes? > > (A.&i.&.|.~ !) 4 > 3 2 1 0 > 3 2 0 1 > 3 1 2 0 > 3 1 0 2 > 3 0 2 1 > 3 0 1 2 > 2 3 1 0 > 2 3 0 1 > 2 1 3 0 > 2 1 0 3 > 2 0 3 1 > 2 0 1 3 > 1 3 2 0 > 1 3 0 2 > 1 2 3 0 > 1 2 0 3 > 1 0 3 2 > 1 0 2 3 > 0 3 2 1 > 0 3 1 2 > 0 2 3 1 > 0 2 1 3 > 0 1 3 2 > 0 1 2 3 > > Thanks, > > -- > Raul > > On Sun, Mar 20, 2022 at 7:34 AM Ian Clark <[email protected]> wrote: > > > > Raul wrote: > > > > > someone with a stronger background in permutations might be able to > describe > > some useful techniques. > > > > > > Ascending Radix numerals (aka the Factorial Number System) offer a > natural > > way of putting permutations in 1-1 correspondence with the positive > > integers. These articles explain how: > > > > [1] https://en.wikipedia.org/wiki/Permutation#Numbering_permutations > > > > [2] https://en.wikipedia.org/wiki/Lehmer_code > > > > [3] https://en.wikipedia.org/wiki/Factorial_number_system > > > > > > Let's define a permutation on 4 points to be equivalent to a > corresponding > > permutation on 5 points which fixes the 5th point. > > > > This is roughly analogous to saying the decimal numerals 3276 and 03276 > > (say) are "equivalent". > > > > EXAMPLE: these permutations are equivalent: > > > > 3 2 1 0 > > > > 3 2 1 0 4 > > > > 3 2 1 0 4 5 > > > > 3 2 1 0 4 5 6 > > > > etc. > > > > > > Essentially (A.) follows the Lehmer approach, but does it back to front. > > Lehmer gives the same integer for two "equivalent" permutations, whereas > > (A.) doesn't… > > > > > > A. 3 2 1 0 > > > > 23 > > > > A. 3 2 1 0 4 > > > > 86 > > > > > > For this reason I feel the methods of [1], [2] and [3] offer a > > better-behaved way of enumerating permutations than (A.). > > > > > > Back in 1970 I worked on fast computer algorithms to do this on a > Burroughs > > B6700 and eventually stumbled on the Lehmer code myself, or something > very > > much like it, which I implemented in Algol. Alas the code is no longer > > extant, but I still have the manual algorithm. I'm going to dust off this > > old work, get my algorithm working as an explicit verb and derive a tacit > > implementation. But someone with a fresher brain could doubtless take [2] > > and implement Lehmer codes efficiently in J far quicker than I shall. > > > > On Mon, 14 Mar 2022 at 18:37, Raul Miller <[email protected]> wrote: > > > > > Watching this, I got to thinking about your anagram index of the "by > suit" > > > deck. > > > > > > We know that an anagram index is in some sense based on the factorial > > > of the length of the sequence. > > > > > > But, also, we can use the indices of the sequence as a base for the > > > anagram index, if we toss the 0 and 1. For example: > > > > > > ((i.!4)A.i.4),._,.(_2}.i.-4)#:i.!4 > > > 0 1 2 3 _ 0 0 > > > 0 1 3 2 _ 0 1 > > > 0 2 1 3 _ 1 0 > > > 0 2 3 1 _ 1 1 > > > 0 3 1 2 _ 2 0 > > > 0 3 2 1 _ 2 1 > > > 1 0 2 3 _ 0 0 > > > 1 0 3 2 _ 0 1 > > > 1 2 0 3 _ 1 0 > > > 1 2 3 0 _ 1 1 > > > 1 3 0 2 _ 2 0 > > > 1 3 2 0 _ 2 1 > > > 2 0 1 3 _ 0 0 > > > 2 0 3 1 _ 0 1 > > > 2 1 0 3 _ 1 0 > > > 2 1 3 0 _ 1 1 > > > 2 3 0 1 _ 2 0 > > > 2 3 1 0 _ 2 1 > > > 3 0 1 2 _ 0 0 > > > 3 0 2 1 _ 0 1 > > > 3 1 0 2 _ 1 0 > > > 3 1 2 0 _ 1 1 > > > 3 2 0 1 _ 2 0 > > > 3 2 1 0 _ 2 1 > > > > > > We can see here that the swaps in the permutations (the numbers on the > > > left) correspond in some way to the "anagram index in base 0 1-.~i.-#" > > > (the numbers on the right). Here, I guess we ignore the leftmost and > > > rightmost values in the permutation, since they are fully determined, > > > in the sense of swaps, by the other values. > > > > > > So, now we have a way of looking at the anagram index of the > > > transposed deck of cards: > > > > > > (_2}.i.-52)#:A.,|:i.4 13 > > > 12 24 36 0 11 22 33 0 10 20 30 0 9 18 27 0 8 16 24 0 7 14 21 0 6 12 18 > > > 0 5 10 15 0 4 8 12 0 3 6 9 0 2 4 6 0 1 2 3 0 0 0 > > > > > > At first, the pattern there doesn't really leap out at us. But, after > > > a short look, it's obvious that these numbers occur in sequences of > > > four. So: > > > > > > _4]\(_2}.i.-52)#:A.,|:i.4 13 > > > 12 24 36 0 > > > 11 22 33 0 > > > 10 20 30 0 > > > 9 18 27 0 > > > 8 16 24 0 > > > 7 14 21 0 > > > 6 12 18 0 > > > 5 10 15 0 > > > 4 8 12 0 > > > 3 6 9 0 > > > 2 4 6 0 > > > 1 2 3 0 > > > > > > In other words, it's: > > > > > > _2}.,(i._13) */ 1|.i.4 > > > 12 24 36 0 11 22 33 0 10 20 30 0 9 18 27 0 8 16 24 0 7 14 21 0 6 12 18 > > > 0 5 10 15 0 4 8 12 0 3 6 9 0 2 4 6 0 1 2 3 0 0 0 > > > > > > Now... I should probably do some more experiments here, to see if I > > > could form a more general insight into how we might compute an > > > arbitrary anagram index. But there's definitely a pattern here, and > > > someone with a stronger background in permutations might be able to > > > describe some useful techniques. > > > > > > Thanks, > > > > > > -- > > > Raul > > > > > > On Sun, Mar 13, 2022 at 5:22 PM Michal Wallace < > [email protected]> > > > wrote: > > > > > > > > I made a short beginner video about dealing cards, using ? and A. : > > > > > > > > https://www.youtube.com/watch?v=eXGKK8BkCkg > > > > > ---------------------------------------------------------------------- > > > > 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 > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
