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A353837
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Number of integer partitions of n with all distinct run-sums.
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42
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1, 1, 2, 3, 4, 7, 10, 14, 17, 28, 35, 49, 62, 85, 107, 149, 174, 238, 305, 384, 476, 614, 752, 950, 1148, 1451, 1763, 2205, 2654, 3259, 3966, 4807, 5773, 7039, 8404, 10129, 12140, 14528, 17288, 20668, 24505, 29062, 34437, 40704, 48059, 56748, 66577, 78228
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OFFSET
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0,3
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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). The first partition whose run-sums are not all distinct is (2,1,1).
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LINKS
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EXAMPLE
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The a(0) = 1 through a(6) = 10 partitions:
() (1) (2) (3) (4) (5) (6)
(11) (21) (22) (32) (33)
(111) (31) (41) (42)
(1111) (221) (51)
(311) (222)
(2111) (321)
(11111) (411)
(2211)
(21111)
(111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@Total/@Split[#]&]], {n, 0, 15}]
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PROG
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(Sage) a353837 = lambda n: sum( abs(BipartiteGraph( Matrix(len(p), len(D:=list(set.union(*map(lambda t: set(divisors(t)), p)))), lambda i, j: p[i]%D[j]==0) ).matching_polynomial()[len(D)-len(p)]) for p in Partitions(n, max_slope=-1) ) # Max Alekseyev, Sep 11 2023
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CROSSREFS
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For multiplicities instead of run-sums we have A098859, ranked by A130091.
A005811 counts runs in binary expansion.
A351014 counts distinct runs in standard compositions.
A353832 represents the operation of taking run-sums of a partition.
A353849 counts distinct run-sums in standard compositions.
Cf. A000041, A008284, A047966, A071625, A073093, A116608, A175413, A181819, A333755, A353848, A353867.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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