cp's OEIS Frontend

A317142 Number of refinement-ordered pairs of strict integer partitions of n.

%I A317142 #18 Apr 02 2025 15:16:59
%S A317142 1,1,1,3,3,5,9,12,16,24,37,47,68,90,123,180,228,307,408,540,694,970,
%T A317142 1207,1598,2048,2669,3357,4382,5599,7109,8990,11428,14330,18144,22652,
%U A317142 28343,35746,44269,55094,68384,84780,104477,129360,158682,195323,240177,293704
%N A317142 Number of refinement-ordered pairs of strict integer partitions of n.
%C A317142 If x and y are strict partitions of the same integer and it is possible to produce x by further partitioning the parts of y, flattening, and sorting, then x <= y.
%C A317142 This sequence is dominated by A294617 (set partitions of strict partitions).
%H A317142 Robert Price, <a href="/A317142/b317142.txt">Table of n, a(n) for n = 0..70</a>
%e A317142 The a(9) = 24 refinement-ordered pairs:
%e A317142     (9)<=(9)
%e A317142   (5,4)<=(9)   (5,4)<=(5,4)
%e A317142   (6,3)<=(9)   (6,3)<=(6,3)
%e A317142   (7,2)<=(9)   (7,2)<=(7,2)
%e A317142   (8,1)<=(9)   (8,1)<=(8,1)
%e A317142 (4,3,2)<=(9) (4,3,2)<=(5,4) (4,3,2)<=(6,3) (4,3,2)<=(7,2) (4,3,2)<=(4,3,2)
%e A317142 (5,3,1)<=(9) (5,3,1)<=(5,4) (5,3,1)<=(6,3) (5,3,1)<=(8,1) (5,3,1)<=(5,3,1)
%e A317142 (6,2,1)<=(9) (6,2,1)<=(6,3) (6,2,1)<=(7,2) (6,2,1)<=(8,1) (6,2,1)<=(6,2,1)
%t A317142 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A317142 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A317142 Table[Sum[Length[Union[Select[Sort/@Map[Total,mps[ptn],{2}],UnsameQ@@#&]]],{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]
%Y A317142 Cf. A002846, A108917, A122768, A213427, A265947, A284640, A294617, A299201, A300383, A317141.
%K A317142 nonn
%O A317142 0,4
%A A317142 _Gus Wiseman_, Jul 22 2018