A318915 - OEIS

%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