Note that you can have many proofs for the same identity. Also, I did not download your pdf, because I did not feel like signing into my facebook account. So, for example, I do not know how your identities treat the relationship between the pascal and sierpinski triangles.
-- Raul On Thu, Jun 6, 2013 at 3:37 PM, Bo Jacoby <[email protected]> wrote: > 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 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
