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A355385
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Number of pairs (y, v) of integer partitions of n where the length of v equals the number of distinct parts in y.
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7
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1, 1, 2, 3, 7, 12, 25, 43, 81, 141, 243, 409, 699, 1132, 1844, 2995, 4744, 7408, 11655, 17839, 27509, 41546, 62879, 93537, 139974, 205547, 302714, 440097, 640968, 921774, 1327538, 1891548, 2696635, 3809860, 5380257, 7540778, 10561566, 14687109, 20408170, 28183998, 38882009
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OFFSET
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0,3
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COMMENTS
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Also the number of composable pairs of integer partitions of n, where a partition is regarded as an arrow from (number of parts) to (number of distinct parts). Is there a nice choice of a composition operation making this into an associative category?
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LINKS
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FORMULA
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EXAMPLE
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The a(0) = 1 through a(5) = 10 pairs:
()() (1)(1) (2)(2) (3)(3) (4)(4) (5)(5)
(11)(2) (21)(21) (31)(31) (41)(41)
(111)(3) (31)(22) (41)(32)
(22)(4) (32)(41)
(211)(31) (32)(32)
(211)(22) (311)(41)
(1111)(4) (311)(32)
(221)(41)
(221)(32)
(2111)(41)
(2111)(32)
(11111)(5)
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MATHEMATICA
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Table[Length[Select[Tuples[IntegerPartitions[n], 2], Length[Union[#[[1]]]]==Length[#[[2]]]&]], {n, 0, 15}]
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PROG
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P(n, y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(n, y) = {prod(k=1, n, 1 + y/(1 - x^k) - y + O(x*x^n))}
seq(n) = {my(g=Vec(P(n, y)), h=Vec(R(n, y))); vector(n+1, i, my(p=g[i], q=h[i]); sum(j=0, poldegree(q), polcoef(p, j)*polcoef(q, j)))} \\ Andrew Howroyd, Dec 31 2022
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CROSSREFS
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The inhomogeneous version with containment and multiplicity is A339006.
The inhomogeneous version with containment is A355383.
The inhomogeneous version with containment for compositions is A355384.
The version for compositions is A355388.
A001970 counts multiset partitions of partitions.
A063834 counts partitions of each part of a partition.
A323583 counts splittings of partitions.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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