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A325852
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Number of (strict) integer partitions of n whose differences of all degrees are nonzero.
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8
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1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 11, 15, 19, 19, 26, 31, 31, 41, 49, 53, 62, 75, 81, 97, 112, 124, 145, 171, 175, 215, 244, 274, 307, 344, 388, 446, 497, 561, 599, 700, 779, 881, 981, 1054, 1184, 1340, 1500, 1669, 1767, 2031, 2237, 2486, 2765, 2946, 3300
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OFFSET
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0,4
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COMMENTS
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The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(11) = 11 partitions (A = 10, B = 11):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)
(21) (31) (32) (42) (43) (53) (54) (64) (65)
(41) (51) (52) (62) (63) (73) (74)
(61) (71) (72) (82) (83)
(421) (431) (81) (91) (92)
(521) (621) (532) (A1)
(541) (542)
(631) (632)
(721) (641)
(731)
(821)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], !MemberQ[Union@@Table[Differences[#, i], {i, Length[#]}], 0]&]], {n, 0, 30}]
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CROSSREFS
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The case for only degrees > 1 is A325874.
Cf. A049988, A175342, A238423, A279945, A295370, A325328, A325468, A325545, A325850, A325851, A325875.
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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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