Yes, Roger, but if you exclude the special cases, the remaining number of identities to learn is, practically speaking, finite. I have a collection of identities in http://www.academia.edu/3247833/Statistical_induction_and_prediction to supplement those in Concrete Mathematics.
- Bo >________________________________ > Fra: Roger Hui <[email protected]> >Til: Programming forum <[email protected]> >Sendt: 18:41 torsdag den 6. juni 2013 >Emne: Re: [Jprogramming] Finding repeated substrings > > >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 > > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
