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

0,3

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

Table of n, a(n) for n=0..30.

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

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

Sequence in context: A092929 A138879 A336136 * A322439 A018313 A219039

Adjacent sequences: A318912 A318913 A318914 * A318916 A318917 A318918

Keyword

nonn,more

Author

Gus Wiseman, Sep 05 2018

Extensions

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

Status

approved