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Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.
11

%I #6 May 03 2019 08:35:25

%S 1,1,2,3,5,6,10,11,15,19,24,25,36,37,43,54,63,64,80,81,100,113,122,

%T 123,151,166,178,195,217,218,269,270,295,316,332,372,424,425,447,472,

%U 547,550,616,617,659,750,777,782,862,885,995,1032,1083,1090,1176,1275

%N Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.

%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 A325400.

%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) = 15 reversed partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (12) (13) (14) (15) (16) (17)

%e (111) (22) (23) (24) (25) (26)

%e (112) (113) (33) (34) (35)

%e (1111) (1112) (114) (115) (44)

%e (11111) (123) (124) (116)

%e (222) (223) (125)

%e (1113) (1114) (224)

%e (11112) (11113) (1115)

%e (111111) (111112) (1124)

%e (1111111) (2222)

%e (11114)

%e (111113)

%e (1111112)

%e (11111111)

%t Table[Length[Select[Sort/@IntegerPartitions[n],And@@Table[OrderedQ[Differences[#,k]],{k,0,Length[#]}]&]],{n,0,30}]

%Y Cf. A007294, A240026, A325353, A325356, A325360, A325362, A325391, A325393, A325394, A325400, A325404, A325406, A325468.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 02 2019