%I #33 Jul 08 2024 12:50:48
%S 1,1,3,5,11,15,33,41,77,105,173,215,381,449,699,911,1335,1611,2433,
%T 2867,4179,5113,6903,8251,11769,13661,18177,22011,28997,33711,45251
%N Number of joining pairs of integer partitions of n.
%C Two integer partitions are a joining pair if they have no common cover (coarser partition) other than the maximum. For example, (221) and (311) are not a joining pair as they are both covered by (32) or (41), while (222) and (33) are a joining pair.
%C All terms are odd.
%C The same as the number of pairs of integer partitions of n without common subsums. - _Mamuka Jibladze_, Jun 16 2024
%H P. Erdős, J. Nicolas and A. Sárközy, <a href="https://eudml.org/doc/210135">On the number of pairs of partitions of n without common subsums</a>, Colloquium Mathematicae, 63 (1992), 61-83.
%F a(n) >= 2 * A000041(n) - 1. - _Alois P. Heinz_, Sep 06 2018
%e The sequence of joining pairs of integer partitions begins:
%e ()() (1)(1) (2)(2) (3)(3) (4)(4) (5)(5)
%e (2)(11) (3)(21) (4)(31) (5)(41)
%e (11)(2) (3)(111) (4)(22) (5)(32)
%e (21)(3) (4)(211) (5)(311)
%e (111)(3) (4)(1111) (5)(221)
%e (31)(4) (5)(2111)
%e (31)(22) (5)(11111)
%e (22)(4) (41)(5)
%e (22)(31) (41)(32)
%e (211)(4) (32)(5)
%e (1111)(4) (32)(41)
%e (311)(5)
%e (221)(5)
%e (2111)(5)
%e (11111)(5)
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t ptncaps[y_]:=Union[Map[Sort[Total/@#,Greater]&,mps[y],{1}]];
%t Table[Select[Tuples[IntegerPartitions[n],2],Intersection@@ptncaps/@#=={{n}}&]//Length,{n,6}]
%Y Cf. A000041, A059849, A060639, A181939, A265947, A299925, A300383, A317141, A317143.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Sep 05 2018
%E a(13)-a(30) from _Alois P. Heinz_, Sep 05 2018