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

1, 1, 1, 3, 3, 5, 9, 12, 16, 24, 37, 47, 68, 90, 123, 180, 228, 307, 408, 540, 694, 970, 1207, 1598, 2048, 2669, 3357, 4382, 5599, 7109, 8990, 11428, 14330, 18144, 22652, 28343, 35746, 44269, 55094, 68384, 84780, 104477, 129360, 158682, 195323, 240177, 293704

Offset: 0

Author

Gus Wiseman, Jul 22 2018

Keywords

• nonn

Comments

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.

This sequence is dominated by A294617 (set partitions of strict partitions).

Examples

			The a(9) = 24 refinement-ordered pairs:
    (9)<=(9)
  (5,4)<=(9)   (5,4)<=(5,4)
  (6,3)<=(9)   (6,3)<=(6,3)
  (7,2)<=(9)   (7,2)<=(7,2)
  (8,1)<=(9)   (8,1)<=(8,1)
(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)
(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)
(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)
		

Links

• Robert Price, Table of n, a(n) for n = 0..70

Crossrefs

Cf. A002846, A108917, A122768, A213427, A265947, A284640, A294617, A299201, A300383, A317141.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[Length[Union[Select[Sort/@Map[Total,mps[ptn],{2}],UnsameQ@@#&]]],{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]