A quick google of "Hockey Stick Theorem" (aka Christmas Stocking Theorem) indicates that the butt end of the stick always start at an edge of the Pascal's Triangle. As explained in the J wiki essay on Pascal's Ladder, the diagonal line can start anywhere in the triangle and the property remains true.
On Thu, Jun 6, 2013 at 9:41 AM, Roger Hui <[email protected]> wrote: > 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
