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A239319
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Irregular triangular array: t(n,k) = number of times that the k-th condensed partition (in Mathematica order) of n occurs in the list of all the condensed partitions of n.
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1
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1, 2, 2, 1, 3, 1, 1, 2, 2, 3, 4, 1, 4, 1, 1, 2, 3, 3, 5, 1, 1, 4, 1, 6, 3, 1, 2, 3, 1, 1, 3, 3, 3, 6, 2, 5, 1, 1, 5, 1, 4, 2, 6, 3, 1, 10, 2, 2, 2, 4, 1, 1, 2, 2, 2, 3, 5, 7, 2, 5, 1, 1, 7, 5, 6, 5, 1, 1, 4, 1, 6, 1, 6, 4, 2, 9, 3, 2, 3, 2, 4, 3, 10, 6, 1, 2
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OFFSET
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1,2
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COMMENTS
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Suppose that p is a partition of n. Let x(1), x(2), ..., x(k) be the distinct parts of p, and let m(i) be the multiplicity of x(i) in p. Let c(p) be the partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n. As defined at A239312, a partition q of n is a condensed partition n if q = f(p) for some partition p of n. The number of numbers in row n is A239312(n); (sum of numbers in row n) = A000041(n), and column 1 is given by A000005.
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LINKS
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EXAMPLE
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The partitions of 6 are 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111, with respective condensations 6, 51, 42, 42, 6, 321, 33, 6, 42, 42, 6. There are A239312(6) = 5 of these, listed here in Mathematica order (x(1) >= x(2) >= ... ): 6, 51, 42, 33, 321. These occur in the list of condensed partitions with multiplicities 4, 1, 4, 1, 1, which is row 6 of the array. First 9 rows:
1
2
2 1
3 1 1
2 2 3
4 1 4 1 1
2 3 3 5 1 1
4 1 6 3 1 2 3 1 1
3 3 3 6 2 5 1 1 5 1
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MATHEMATICA
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z = 15; u[n_, k_] := u[n, k] = Map[Total, Split[IntegerPartitions[n][[k]]]]; r[n_] := r[n] = Table[Reverse[Sort[u[n, k]]], {k, 1, PartitionsP[n]}] ; t[n_] := t[n] = DeleteDuplicates[r[n]]; u[n_] := u[n] = Length[t[n]] ; v = Table[Count[r[n], t[n][[k]]], {n, 1, z}, {k, 1, u[n]}]; TableForm[v] (* A239319, array *)
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CROSSREFS
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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_Clark Kimberling_, Mar 15 2014
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STATUS
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approved
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