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%I #6 May 03 2019 08:35:16
%S 1,1,2,3,4,5,7,7,9,11,12,13,17,16,19,23,23,24,30,29,35,37,37,40,49,47,
%T 51,56,59,61,73,65,75,80,84,91,99,91,103,112,120,114,132,126,143,154,
%U 147,152,175,169,190,187,194,198,226,225,231,236,246,256,293
%N Number of integer partitions of n whose k-th differences are weakly decreasing for all k >= 0.
%C 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).
%C The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
%C The Heinz numbers of these partitions are given by A325397.
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e The a(1) = 1 through a(8) = 9 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (21) (22) (32) (33) (43) (44)
%e (111) (31) (41) (42) (52) (53)
%e (1111) (221) (51) (61) (62)
%e (11111) (222) (331) (71)
%e (321) (2221) (332)
%e (111111) (1111111) (431)
%e (2222)
%e (11111111)
%e The first partition that has weakly decreasing differences (A320466) but is not counted under a(9) is (3,3,2,1), whose first and second differences are (0,-1,-1) and (-1,0) respectively.
%t Table[Length[Select[IntegerPartitions[n],And@@Table[GreaterEqual@@Differences[#,k],{k,0,Length[#]}]&]],{n,0,30}]
%Y Cf. A320466, A320509, A325350, A325354, A325391, A325393, A325397, A325398, A325399, A325404, A325405, A325406, A325468.
%K nonn
%O 0,3
%A _Gus Wiseman_, May 02 2019
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