Partitions

• To: mathgroup at smc.vnet.net
• Subject: [mg32852] Partitions
• From: "Juan" <erfa11 at hotmail.com>
• Date: Fri, 15 Feb 2002 02:49:56 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Hello,
I sent this message four days ago, but it have not appeared yet in our
MathGroup; sure I did something wrong. I try again.

If you want the partitions of n, in s parts with different numbers, here is
a way to do it:

In[1]:= << "DiscreteMath`Combinatorica`"

In[2]:= F1[n_, s_] := Select[Partitions[n], s == Length[#1] ==
Length[Union[#1]] & ]

In[4]:= Timing[F1[50, 8]; ]
Out[4]= {77.631*Second, Null}

I am written a packade to play lottery and I need that for n = 100, 200,
300,...
So, using the brute force philosophy, I wrote the F2  function:

In[5]:= F2[n_, s_, m_:1] :=
Module[{a = Array[p, s - 1], y, v, q, t, b = Abs[s - 2]},
p[0] = m - 1;
y = FoldList[Plus, 0, Range[n - 1]];
v = Flatten[{a, n - Plus @@ a}];
q = n - FoldList[Plus, 0, a];
t = Table[{p[i + 1], p[i] + 1, (q[[i + 1]] - y[[s - i]])/(s - i)}, {i,
0, s - 2}];
Flatten[Table[v, Evaluate[Sequence @@ t]], b]]

In[6]:= Timing[F2[50, 8]; ]
Out[6]= {0.030000000000001137*Second, Null}

In[7]:=F1[15, 4]
Out[7]={{9, 3, 2, 1}, {8, 4, 2, 1}, {7, 5, 2, 1}, {7, 4, 3, 1}, {6, 5, 3,
1}, {6, 4, 3, 2}}
In[8]:=F2[15, 4]
Out[8]={{1, 2, 3, 9}, {1, 2, 4, 8}, {1, 2, 5, 7}, {1, 3, 4, 7}, {1, 3, 5,
6}, {2, 3, 4, 6}}

In[9]:=Length[F1[40, 7]] == Length[F2[40, 7]]
Out[9]=True

In[10]:= Timing[F2[100, 10]; ]
Out[10]= {7.170999999999992*Second, Null}

Surely F2 can be improved, or If you know something better, please send it
here.

About the parameters of F2[n, s, m],
n and s are integers positives and s<= (Sqrt[1+8 n]-1)/2, m is a integer.

Thanks
Juan

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```

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