OFFSET
0,3
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
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.
FORMULA
a(n) >= 2 * A000041(n) - 1. - Alois P. Heinz, Sep 06 2018
EXAMPLE
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)
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}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 05 2018
EXTENSIONS
a(13)-a(30) from Alois P. Heinz, Sep 05 2018
STATUS
approved