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%I #28 May 25 2019 01:54:42
%S 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,
%T 66,92,135,203,15,30,47,64,57,98,141,109,137,198,296,249,371,566,877,
%U 22,45,77,105,97,171,250,109,212,269,392,592,300,444,560,850,1315,712,1075
%N Number of partitions of a multiset whose signature is the n-th partition (in Mathematica order).
%C 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:
%C n.....#elements.....n-th partition
%C 0.....0 elements:....[]
%C 1.....1 element:.....[1]
%C 2.....2 elements:....[2]
%C 3....................[1,1]
%C 4.....3 elements:....[3]
%C 5....................[2,1]
%C 6....................[1,1,1]
%C 7.....4 elements:....[4]
%C 8....................[3,1]
%C 9....................[2,2]
%C 10...................[2,1,1]
%C 11...................[1,1,1,1]
%C 12....5 elements:....[5]
%C 13...................[4,1]
%C A000041 and A000110 are subsequences for conjugate partitions. A000070 and A035098 are also subsequences for conjugate partitions. - _Alford Arnold_, Dec 31 2005
%C A002774 and A020555 is another pair of subsequences for conjugate partitions. - _Franklin T. Adams-Watters_, May 16 2006
%H Jun Kyo Kim and Sang Guen Hahn, <a href="http://www.emis.de/journals/HOA/IJMMS/22/1213.pdf">Recursive Formulae for the Multiplicative Partition Function</a>, Internat. J. Math. & Math. Sci., 22(1) (1999), 213-216.
%H A. Knopfmacher, M. E. Mays, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.113.7323">A survey of factorization counting functions</a>, International Journal of Number Theory, 1(4):563-581,(2005). See P(n) page 3.
%e 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:
%e {{a,a,b,c}}
%e {{a,a,b},{c}}
%e {{a,a,c},{b}}
%e {{a,b,c},{a}}
%e {{a,a},{b,c}}
%e {{a,b},{a,c}}
%e {{a,a},{b},{c}}
%e {{a,b},{a},{c}}
%e {{a,c},{a},{b}}
%e {{b,c},{a},{a}}
%e {{a},{a},{b},{c}}
%e We see there are 11 partitions of this multiset, so a(10)=11.
%e 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.
%t MultiPartiteP[n : {___Integer?NonNegative}] :=
%t Block[{p, $RecursionLimit = 1024, firstPositive},
%t firstPositive =
%t Compile[{{vv, _Integer, 1}},
%t Module[{k = 1}, Do[If[el == 0, k++, Break[]], {el, vv}]; k]];
%t p[{0 ...}] := 1;
%t p[v_] :=
%t p[v] = Module[{len = Length[v], it, k, zeros, sum, pos, gcd},
%t it = Array[k, len];
%t pos = firstPositive[v];
%t zeros = ConstantArray[0, len];
%t sum = 0;
%t Do[If[it == zeros, Continue[]];
%t gcd = GCD @@ it;
%t sum += it[[pos]] DivisorSigma[-1, gcd] p[v - it];,
%t Evaluate[Sequence @@ Thread[{it, 0, v}]]];
%t sum/v[[pos]]];
%t p[n]];
%t ParallelMap[MultiPartiteP,
%t Flatten[Table[IntegerPartitions[k], {k, 0, 8}], 1]]
%t (* Oleksandr Pavlyk, Jan 23 2011 *)
%Y Cf. A035098, A035310.
%K nonn
%O 0,3
%A _Jon Wild_, Aug 11 2004
%E Edited by _Franklin T. Adams-Watters_, May 16 2006
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