To quote Graham, Knuth, & Patashnik, *Concrete Mathematics*, page 155: The numbers in Pascal's triangle satisfy, practically speaking, infinitely many identities, so it's not too surprising that we can find some surprising relationships by looking closely.
The relationship you quoted, (>:x)!y ←→ +/x!i.y, can be generalized into a theorem that I called Pascal's Ladder<http://www.jsoftware.com/jwiki/Essays/Pascal%27s%20Ladder> (I like that better than "Hockey Stick Theorem"). On Thu, Jun 6, 2013 at 9:18 AM, Bo Jacoby <[email protected]> wrote: > The theorem that 2!y is equal to +/i.y is a special case of the more > general theorem that (>:x)!y is equal to +/x!i.y > > - Bo > > > >________________________________ > > Fra: Roger Hui <[email protected]> > >Til: Programming forum <[email protected]> > >Sendt: 16:41 torsdag den 6. juni 2013 > >Emne: Re: [Jprogramming] Finding repeated substrings > > > > > >There is a proof of a very similar theorem in section 1.4 of *Notation as > a > >Tool of Thought <http://www.jsoftware.com/papers/tot.htm>*. (The > >difference is that index origin is 1 in the paper.) > > > > > > > >On Thu, Jun 6, 2013 at 6:44 AM, Raul Miller <[email protected]> > wrote: > > > >> Caution: this code can give an incomplete result. For example, I do > >> not believe it will find 'aabaab'. Rather than fix this, I'll defer to > >> other solutions in this thread (which I imagine properly address this > >> issue). > >> > >> If anyone wants to take this code and fix it, the first instance of 2 > >> -~/\ ] should be replaced with a mechanism that treats all > >> combinations of 2 (and not just adjacent pairs). > >> > >> (And on that note, I Tracy Harms recently directed my attention to a > >> page with a beautiful proof that 2&! is +/@i. - that concept would be > >> useful, here, I think. I wish I had recorded the url of that page. But > >> the gist of my thought is that it should be possible to go from y and > >> a member of i.2!y to a unique pair of two numbers in the range i.y, > >> and that might be a nice way of implementing this "combinations of 2" > >> function.) > >> > >> FYI, > >> > >> -- > >> Raul > >> > >---------------------------------------------------------------------- > >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
