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A325350
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Number of integer partitions of n whose augmented differences are weakly decreasing.
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9
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1, 1, 2, 3, 4, 6, 8, 10, 13, 17, 21, 26, 32, 38, 46, 56, 66, 78, 92, 106, 124, 145, 166, 191, 220, 249, 284, 325, 366, 413, 468, 523, 586, 659, 733, 817, 913, 1011, 1121, 1245, 1373, 1515, 1674, 1838, 2020, 2223, 2433, 2664, 2920, 3184, 3476, 3797, 4129, 4492
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OFFSET
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0,3
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COMMENTS
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The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The Heinz numbers of these partitions are given by A325389.
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LINKS
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FORMULA
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G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - Ilya Gutkovskiy, Apr 25 2019
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EXAMPLE
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The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (31) (32) (42) (52) (53)
(111) (211) (41) (51) (61) (62)
(1111) (311) (321) (421) (71)
(2111) (411) (511) (521)
(11111) (3111) (3211) (611)
(21111) (4111) (4211)
(111111) (31111) (5111)
(211111) (32111)
(1111111) (41111)
(311111)
(2111111)
(11111111)
For example, (4,2,1,1) has augmented differences (3,2,1,1), which are weakly decreasing, so (4,2,1,1) is counted under a(8).
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MATHEMATICA
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aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], OrderedQ[Reverse[aug[#]]]&]], {n, 0, 30}]
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CROSSREFS
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Cf. A007294, A098859, A240026, A320466, A320509, A325349, A325353, A325354, A325356, A325357, A325358, A325361, A325364.
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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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