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A353843
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Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory ending in a partition of length k. All zeros removed.
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3
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1, 1, 2, 2, 1, 4, 1, 2, 5, 5, 5, 1, 2, 12, 1, 8, 11, 3, 3, 19, 8, 5, 27, 9, 1, 2, 34, 19, 1, 15, 26, 34, 2, 2, 49, 45, 5, 5, 68, 48, 14, 4, 58, 98, 15, 1, 18, 76, 105, 31, 1, 2, 88, 159, 46, 2, 13, 98, 191, 79, 4, 2, 114, 261, 105, 8, 14, 148, 282, 164, 19
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OFFSET
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0,3
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COMMENTS
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The partition run-sum trajectory is obtained by repeatedly taking the run-sums until a strict partition is reached. For example, the trajectory of y = (3,2,1,1,1) is (3,2,1,1,1) -> (3,3,2) -> (6,2), so y is counted under T(8,2).
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LINKS
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EXAMPLE
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Triangle begins:
1
1
2
2 1
4 1
2 5
5 5 1
2 12 1
8 11 3
3 19 8
5 27 9 1
2 34 19 1
15 26 34 2
2 49 45 5
5 68 48 14
4 58 98 15 1
For example, row n = 8 counts the following partitions:
(8) (53) (431)
(44) (62) (521)
(422) (71) (3221)
(2222) (332)
(4211) (611)
(41111) (3311)
(221111) (5111)
(11111111) (22211)
(32111)
(311111)
(2111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Length[FixedPoint[Sort[Total/@Split[#]]&, #]]==k&]], {n, 0, 15}, {k, 0, n}]
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CROSSREFS
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The last part of the same trajectory is A353842.
The length of the trajectory is A353846.
The version for compositions is A353856.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353932 lists run-sums of standard compositions.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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