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Number of strict multiset partitions of multiset partitions of integer partitions of n.
1

%I #8 Jul 18 2021 20:55:12

%S 1,1,3,9,23,62,161,410,1031,2579,6359,15575,37830,91241,218581,520544,

%T 1232431,2902644,6802178,15866054,36844016,85202436,196251933,

%U 450341874,1029709478,2346409350,5329371142,12066816905,27240224766,61317231288,137643961196

%N Number of strict multiset partitions of multiset partitions of integer partitions of n.

%C Number of sets of nonempty multisets of nonempty multisets of positive integers with total sum n.

%H Alois P. Heinz, <a href="/A330453/b330453.txt">Table of n, a(n) for n = 0..3853</a>

%F Weigh transform of A001970. The weigh transform of a sequence (s_1, s_2, ...) is the sequence with generating function Product_{i > 0} (1 + x^i)^s_i.

%e The a(4) = 23 partitions:

%e ((4)) ((22)) ((31)) ((211)) ((1111))

%e ((2)(2)) ((1)(3)) ((1)(21)) ((1)(111))

%e ((1))((3)) ((2)(11)) ((11)(11))

%e ((1)(1)(2)) ((1))((111))

%e ((1))((21)) ((1)(1)(11))

%e ((2))((11)) ((1))((1)(11))

%e ((1))((1)(2)) ((1)(1)(1)(1))

%e ((2))((1)(1)) ((11))((1)(1))

%e ((1))((1)(1)(1))

%p with(numtheory): with(combinat):

%p b:= proc(n) option remember; `if`(n=0, 1, add(add(d*

%p numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)

%p end:

%p a:= proc(n) a(n):= `if`(n<2, 1, add(a(n-k)*add(b(d)

%p *d*(-1)^(k/d+1), d=divisors(k)), k=1..n)/n)

%p end:

%p seq(a(n), n=0..32); # _Alois P. Heinz_, Jul 18 2021

%t ppl[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Union[Sort/@Tuples[ppl[#,k-1]&/@ptn]],{ptn,IntegerPartitions[n]}]];

%t Table[Length[Select[ppl[n,3],UnsameQ@@#&]],{n,0,10}]

%Y The not necessarily strict case is A007713.

%Y Cf. A001055, A001970, A050336, A050343, A089259, A261049, A271619, A316980, A318566, A323787-A323795, A330452-A330459, A330461, A330463.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 17 2019