Re: rules on vector sequence

*To*: mathgroup at smc.vnet.net*Subject*: [mg98070] Re: rules on vector sequence*From*: Bill Rowe <readnews at sbcglobal.net>*Date*: Sun, 29 Mar 2009 02:46:30 -0500 (EST)

On 3/28/09 at 5:41 AM, sottosupremo at gmail.com (Filippo Miatto) wrote: >Dear all, I need to generate all the vectors, of a given length N, >that have these three properties: >1- the entries have to be among -2,-1,0,1,2 >2- the sum of all the entries has to be 0 >3- only two or four of the entries can be different from 0 >do you have any suggestions on how i can do that? i tried something >but without success.. expecially i don't know how to implement the >third rule.. thank you in advance! Filippo Rule 1 lists 5 distinct values. So, each vector can be thought of as a N digit number base 5. And for N = 5, Range[5^4, 5^5 - 1] will be a list of all possible vectors of length 5 encoded as a number base 5. I can implement rule 3 as: test = Pick[Range[5^4, 5^5 - 1], Total[Unitize[IntegerDigits[#, 5] - 2]] & /@ Range[5^4, 5^5 - 1], 2 | 4]; IntegerDigits[value, 5] - 2 converts each encoded vector to the form specified by rule 1. Unitize sets each non-zero value to 1. And Pick chooses just those values with either 2 or 4 non-zero values. Rule 2 can be implemented as: Cases[test, _?(Total[IntegerDigits[#, 5]] == 10 &)] Here, I look for a total of 10 rather than 0 since the digits for base 5 numbers run from 0 to 4. There are a total of In[33]:= Length@Cases[test, _?(Total[IntegerDigits[#, 5]] == 10 &)] Out[33]= 180 vectors of length 5 meeting your conditions above. The first 5 are: In[35]:= (IntegerDigits[#, 5] - 2) & /@ Cases[test, _?(Total[IntegerDigits[#, 5]] == 10 &)][[;; 5]] Out[35]= {{-1, -2, 0, 1, 2}, {-1, -2, 0, 2, 1}, {-1, -2, 1, 0, 2}, {-1, -2, 1, 2, 0}, {-1, -2, 2, 0, 1}} I'll leave it to you to generalize the above for arbitrary length vectors. Also, note I've not tried to optimize any of this code. If N gets large, this code won't run all that fast as things like IntegerDigits are being done more than once for the same values.