A318915 Number of joining pairs of integer partitions of n.

1, 1, 3, 5, 11, 15, 33, 41, 77, 105, 173, 215, 381, 449, 699, 911, 1335, 1611, 2433, 2867, 4179, 5113, 6903, 8251, 11769, 13661, 18177, 22011, 28997, 33711, 45251

Offset: 0

Author

Gus Wiseman, Sep 05 2018

Keywords

• nonn
• more

Comments

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.

All terms are odd.

The same as the number of pairs of integer partitions of n without common subsums. - Mamuka Jibladze, Jun 16 2024

Examples

			The sequence of joining pairs of integer partitions begins:
  ()()   (1)(1)   (2)(2)    (3)(3)     (4)(4)      (5)(5)
                  (2)(11)   (3)(21)    (4)(31)     (5)(41)
                  (11)(2)   (3)(111)   (4)(22)     (5)(32)
                            (21)(3)    (4)(211)    (5)(311)
                            (111)(3)   (4)(1111)   (5)(221)
                                       (31)(4)     (5)(2111)
                                       (31)(22)    (5)(11111)
                                       (22)(4)     (41)(5)
                                       (22)(31)    (41)(32)
                                       (211)(4)    (32)(5)
                                       (1111)(4)   (32)(41)
                                                   (311)(5)
                                                   (221)(5)
                                                   (2111)(5)
                                                   (11111)(5)
		

Links

• P. Erdős, J. Nicolas and A. Sárközy, On the number of pairs of partitions of n without common subsums, Colloquium Mathematicae, 63 (1992), 61-83.

Crossrefs

Cf. A000041, A059849, A060639, A181939, A265947, A299925, A300383, A317141, A317143.

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]]]];
ptncaps[y_]:=Union[Map[Sort[Total/@#,Greater]&,mps[y],{1}]];
Table[Select[Tuples[IntegerPartitions[n],2],Intersection@@ptncaps/@#=={{n}}&]//Length,{n,6}]

Formula

a(n) >= 2 * A000041(n) - 1. - Alois P. Heinz, Sep 06 2018

Extensions

a(13)-a(30) from Alois P. Heinz, Sep 05 2018