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A279375 Number of set partitions of strict integer partitions of n that have distinct block-sums.

%I A279375 #10 Dec 12 2016 11:27:08
%S A279375 1,1,1,3,3,5,9,12,16,24,39,49,70,94,127,202,247,340,450,606,772,1169,
%T A279375 1407,1920,2454,3267,4089,5469,7293,9222,11884,15291,19417,24890,
%U A279375 31469,39662,52619,64764,82502,103576,131169,162726,206015,254233,318464,406262,499210,620593,773673,957073,1181593
%N A279375 Number of set partitions of strict integer partitions of n that have distinct block-sums.
%C A279375 Also twice partitioned numbers where all partitions are strict. Also triangles of weight n in the multisystem of strict partitions. Strict partitions are an example of a multisystem that is neither transitive nor partitive nor contractible but is decomposable; see link for details.
%H A279375 Alois P. Heinz, <a href="/A279375/b279375.txt">Table of n, a(n) for n = 0..100</a>
%H A279375 Gus Wiseman, <a href="https://docs.google.com/document/d/1m0s6DGTBkDW9gvMuFmJHvy6oLGRAbQ7okAZcOPZawp0/pub">Comcategories and Multiorders</a> <a href="http://www.nafindix.com/math/academic/ComcategoriesandMultiordersv7.pdf">(pdf version)</a>
%e A279375 The a(6)=9 set partitions of strict integer partitions of 6 are: ((6)), ((51)), ((5)(1)), ((42)), ((4)(2)), ((321)), ((32)(1)), ((31)(2)), ((3)(2)(1)). The set partition ((3)(21)) is not counted because its blocks do not have distinct sums.
%t A279375 nn=20;sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A279375 Table[Total[Length[Select[sps[Sort[#]],UnsameQ@@Total/@#&]]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}]
%Y A279375 Cf. A063834, A089259, A258466, A270995, A271619, A275780, A279374.
%K A279375 nonn
%O A279375 0,4
%A A279375 _Gus Wiseman_, Dec 11 2016