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A096443
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Number of partitions of a multiset whose signature is the n-th partition (in Mathematica order).
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14
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1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 11, 15, 7, 12, 16, 21, 26, 36, 52, 11, 19, 29, 38, 31, 52, 74, 66, 92, 135, 203, 15, 30, 47, 64, 57, 98, 141, 109, 137, 198, 296, 249, 371, 566, 877, 22, 45, 77, 105, 97, 171, 250, 109, 212, 269, 392, 592, 300, 444, 560, 850, 1315, 712, 1075
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OFFSET
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0,3
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COMMENTS
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The signature of a multiset is the partition consisting of the multiplicities of its elements; e.g., {a,a,a,b,c} is represented by [3,1,1]. The Mathematica order for partitions orders by ascending number of total elements, then by descending numerical order of its representation. The list begins:
n.....#elements.....n-th partition
0.....0 elements:....[]
1.....1 element:.....[1]
2.....2 elements:....[2]
3....................[1,1]
4.....3 elements:....[3]
5....................[2,1]
6....................[1,1,1]
7.....4 elements:....[4]
8....................[3,1]
9....................[2,2]
10...................[2,1,1]
11...................[1,1,1,1]
12....5 elements:....[5]
13...................[4,1]
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LINKS
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EXAMPLE
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The 10th partition is [2,1,1]. The partitions of a multiset whose elements have multiplicities 2,1,1 - for example, {a,a,b,c} - are:
{{a,a,b,c}}
{{a,a,b},{c}}
{{a,a,c},{b}}
{{a,b,c},{a}}
{{a,a},{b,c}}
{{a,b},{a,c}}
{{a,a},{b},{c}}
{{a,b},{a},{c}}
{{a,c},{a},{b}}
{{b,c},{a},{a}}
{{a},{a},{b},{c}}
We see there are 11 partitions of this multiset, so a(10)=11.
Also, a(n) is the number of distinct factorizations of A063008(n). For example, A063008(10) = 60 and 60 has 11 factorizations: 60, 30*2, 20*3, 15*4, 15*2*2, 12*5, 10*6, 10*3*2, 6*5*2, 5*4*3, 5*3*2*2 which confirms that a(10) = 11.
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MATHEMATICA
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MultiPartiteP[n : {___Integer?NonNegative}] :=
Block[{p, $RecursionLimit = 1024, firstPositive},
firstPositive =
Compile[{{vv, _Integer, 1}},
Module[{k = 1}, Do[If[el == 0, k++, Break[]], {el, vv}]; k]];
p[{0 ...}] := 1;
p[v_] :=
p[v] = Module[{len = Length[v], it, k, zeros, sum, pos, gcd},
it = Array[k, len];
pos = firstPositive[v];
zeros = ConstantArray[0, len];
sum = 0;
Do[If[it == zeros, Continue[]];
gcd = GCD @@ it;
sum += it[[pos]] DivisorSigma[-1, gcd] p[v - it]; ,
Evaluate[Sequence @@ Thread[{it, 0, v}]]];
sum/v[[pos]]];
p[n]];
ParallelMap[MultiPartiteP,
Flatten[Table[IntegerPartitions[k], {k, 0, 8}], 1]]
(* Oleksandr Pavlyk, Jan 23 2011 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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