Re: Re: [Q] Generalized Partitions

*To*: mathgroup at smc.vnet.net*Subject*: [mg30047] Re: [mg29993] Re: [mg29942] [Q] Generalized Partitions*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Wed, 25 Jul 2001 01:00:30 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

While thinking of Janusz Kawczak's partition problem I found the following amusing way to compute the number of partitions of an integer n as a sum of k-elements of a given list l. kpartitions[n_, k_, l_List] := Coefficient[ Normal[Series[Times @@ ((1/(1 - t*x^#)) & /@ l), {x, 0, n}, {t, 0, k}]], x^n t^k] I leave the proof that this is correct as an exercise for the reader :) For example: for n=51, k=6 and L={1,4,7,10,13,16,19,22,25,28,31,34,37,40,43,47,50,53,56,59}; In[24]:= kpartitions[51,6,L]//Timing Out[24]= {3.58 Second,109} Unfortunately this is rather slow since on the same computer my Backtrack method gives: In[28]:= Length[sols=Backtrack[sp,partialQ,solutionQ,All]]//Timing Out[28]= {1.01 Second,109} and the even faster iterative approach of Daniel Lichtblau gives: In[32]:= Length[gPartitions[L,51,6]]//Timing Out[32]= {0.32 Second,109} Sadly the solutions that seem mathematically most satisfying are not always the most efficient :( -- Andrzej Kozlowski Toyama International University JAPAN http://platon.c.u-tokyo.ac.jp/andrzej/