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A318915
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Number of joining pairs of integer partitions of n.
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2
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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)
(32)(41)
(221)(5)
(2111)(5)
(11111)(5)
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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