This does help, thanks! I have some questions though. Again I'm a total computer programming novice so I don't understand the code you have written. How would I add the extra conditions to what you have written? I will be looking forward to hearing about the multiset partition support you are working on. I understand the syntax for adding assumptions but from what you have written I'm not sure what variable (?) I would use in an assumption. thanks sonny
On Dec 14, 5:09 pm, "Mike Hansen" <mhan...@gmail.com> wrote: > Hello, > > On Sun, Dec 14, 2008 at 4:04 PM, green351 <mmamashra...@gmail.com> wrote: > > > Hi, > > This is my first time emailing with a question and my first time > > trying to use Sage (I'm a complete programming dunce). I'm trying to > > do the following: > > Given the tuple (p,q) in Z x Z and integer n I need to count the > > number of integer-tuple solutions to the following > > (p,q)=(a_1,b_1)+...+(a_n,b_n) subject to the following conditions > > - (a_i,b_i) \neq (a_j,b_j) for i, j different > > - a_i, b_i >= -1 > > - a_i+b_i > 0 > > What you're trying to count is the number of multiset partitions. I > haven't written any code to do this in Sage, but I've been meaning > too. So now is a good time to start :-) > > For an interesting reference on this is Knuth's Art of Computer > Programming Volume 4, Fascicle 3 (b). Using the formula found > inhttp://www.emis.de/journals/HOA/IJMMS/22/1213.pdf, we get the > following bit of Sage code > > @cached_function > def p(*n): > r = len(n) > s = sum(n) > if s == 0: > return 1 > result = 0 > for k in range(1, s+1): > for l in IntegerVectors(k, r, outer=n): > nml = tuple(a-b for a,b in zip(n,l)) > g = gcd(l) > result += sigma(g)/g*sum(l)*p(*nml) > return result / s > > You call it just like p(a,b). We can duplicate Table A-1 in the Knuth > fascicle: > > sage: [ [p(i,j) for j in range(7)] for i in range(6)] > > [[1, 1, 2, 3, 5, 7, 11], > [1, 2, 4, 7, 12, 19, 30], > [2, 4, 9, 16, 29, 47, 77], > [3, 7, 16, 31, 57, 97, 162], > [5, 12, 29, 57, 109, 189, 323], > [7, 19, 47, 97, 189, 339, 589]] > > We can also get the specific values cited in the paper with the formula: > > sage: p(10,5) > 3804 > sage: p(9,8) > 13715 > sage: p(10,8) > 21893 > sage: p(4,1,1) > 38 > > We can also use this to count the number of distinct factorizations of > an integer. For example, we can write 12 as 12*1, 6*2, 4*3, or 3*2*2. > > sage: p(*[exponent for prime,exponent in factor(12)]) > 4 > > I'll try to add native support for multiset partitions here into Sage > in the near future. > > Hope that helps, > > --Mike --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
