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A353836
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Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct run-sums.
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15
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1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 1, 0, 0, 0, 2, 5, 0, 0, 0, 0, 5, 5, 1, 0, 0, 0, 0, 2, 12, 1, 0, 0, 0, 0, 0, 7, 12, 3, 0, 0, 0, 0, 0, 0, 3, 19, 8, 0, 0, 0, 0, 0, 0, 0, 5, 27, 9, 1, 0, 0, 0, 0, 0, 0, 0, 2, 33, 20, 1, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,5
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COMMENTS
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The run-sums of a sequence are the sums of its maximal consecutive constant subsequences (runs). For example, the run-sums of (2,2,1,1,1,3,2,2) are (4,3,3,4).
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LINKS
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EXAMPLE
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Triangle begins:
1
0 1
0 2 0
0 2 1 0
0 4 1 0 0
0 2 5 0 0 0
0 5 5 1 0 0 0
0 2 12 1 0 0 0 0
0 7 12 3 0 0 0 0 0
0 3 19 8 0 0 0 0 0 0
0 5 27 9 1 0 0 0 0 0 0
0 2 33 20 1 0 0 0 0 0 0 0
0 13 28 34 2 0 0 0 0 0 0 0 0
0 2 48 46 5 0 0 0 0 0 0 0 0 0
0 5 65 51 14 0 0 0 0 0 0 0 0 0 0
0 4 57 99 15 1 0 0 0 0 0 0 0 0 0 0
For example, row n = 8 counts the following partitions:
(8) (53) (431)
(44) (62) (521)
(422) (71) (3221)
(2222) (332)
(41111) (611)
(221111) (3311)
(11111111) (4211)
(5111)
(22211)
(32111)
(311111)
(2111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Split[#]]]==k&]], {n, 0, 15}, {k, 0, n}]
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CROSSREFS
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Counting distinct parts instead of run-sums gives A116608.
A351014 counts distinct runs in standard compositions.
A353832 represents the operation of taking run-sums of a partition.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
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KEYWORD
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AUTHOR
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STATUS
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approved
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