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A375133
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Number of integer partitions of n whose maximal anti-runs have distinct maxima.
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20
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1, 1, 1, 2, 3, 4, 5, 8, 10, 14, 17, 23, 29, 38, 47, 60, 74, 93, 113, 141, 171, 211, 253, 309, 370, 447, 532, 639, 758, 904, 1066, 1265, 1487, 1754, 2053, 2411, 2813, 3289, 3823, 4454, 5161, 5990, 6920, 8005, 9223, 10634, 12218, 14048, 16101, 18462, 21107
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OFFSET
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0,4
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COMMENTS
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An anti-run is a sequence with no adjacent equal parts.
These are partitions with no part appearing more than twice and greatest part appearing only once.
Also the number of reversed integer partitions of n whose maximal anti-runs have distinct maxima.
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LINKS
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FORMULA
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G.f.: Sum_{i>=0} (x^i * Product_{j=1..i-1} (1-x^(3*j))/(1-x^j)). - John Tyler Rascoe, Aug 21 2024
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EXAMPLE
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The partition y = (6,5,5,4,3,3,2,1) has maximal anti-runs ((6,5),(5,4,3),(3,2,1)), with maxima (6,5,3), so y is counted under a(29).
The a(0) = 1 through a(9) = 14 partitions:
() (1) (2) (3) (4) (5) (6) (7) (8) (9)
(21) (31) (32) (42) (43) (53) (54)
(211) (41) (51) (52) (62) (63)
(311) (321) (61) (71) (72)
(411) (322) (422) (81)
(421) (431) (432)
(511) (521) (522)
(3211) (611) (531)
(3221) (621)
(4211) (711)
(4221)
(4311)
(5211)
(32211)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@Max/@Split[#, UnsameQ]&]], {n, 0, 30}]
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PROG
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(PARI)
A_x(N) = {my(x='x+O('x^N), f=sum(i=0, N, (x^i)*prod(j=1, i-1, (1-x^(3*j))/(1-x^j)))); Vec(f)}
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CROSSREFS
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Includes all strict partitions A000009.
For compositions instead of partitions we have A374761.
The complement for minima instead of maxima is A375404, ranks A375399.
A011782 counts integer compositions.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
Cf. A141199, A279790, A358830, A358833, A358836, A358905, A374704, A374757, A374758, A375136, A375400.
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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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