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Number of twice-partitions of type (Q,P,Q) and weight n.
21

%I #6 Dec 21 2016 11:05:07

%S 1,1,3,3,5,11,12,18,24,49,53,82,102,149,236,297,392,540,702,924,1423,

%T 1723,2318,3016,3969,5037,6647,9285,11448,15048,19143,24695,31288,

%U 40075,50014,68583,83056,107252,133796,171565

%N Number of twice-partitions of type (Q,P,Q) and weight n.

%C Also number of ways to choose a sum-preserving permutation of a set partition of a strict partition of n.

%H Gus Wiseman, <a href="/A063834/a063834.txt">Sequences enumerating triangles of integer partitions</a>

%e The a(6)=11 twice-partitions are:

%e ((6)),

%e ((5)(1)), ((51)),

%e ((4)(2)), ((42)),

%e ((3)(2)(1)), ((3)(21)), ((21)(3)), ((31)(2)), ((32)(1)), ((321)).

%t nn=20;

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Table[Total[Sum[Times@@Factorial/@Length/@Split[Sort[Total/@ptn]],{ptn,sps[Sort[#]]}]&/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,nn}]

%Y Cf. A063834, A279791.

%K nonn

%O 1,3

%A _Gus Wiseman_, Dec 18 2016