A325393 - OEIS

%I #6 May 03 2019 08:36:49

%S 1,1,1,2,2,3,3,4,5,5,6,8,7,9,11,10,12,15,13,16,19,18,20,24,22,26,29,

%T 28,31,37,33,38,43,42,44,52,48,55,59,58,62,72,65,74,80,80,82,94,88,99,

%U 103,104,108,123,114,126,133,135,137,155,145,161,166,169,174

%N Number of integer partitions of n whose k-th differences are strictly 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 A325399.

%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(9) = 5 partitions:

%e   (1)  (2)  (3)   (4)   (5)   (6)   (7)   (8)    (9)

%e             (21)  (31)  (32)  (42)  (43)  (53)   (54)

%e                         (41)  (51)  (52)  (62)   (63)

%e                                     (61)  (71)   (72)

%e                                           (431)  (81)

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

%Y Cf. A049988, A320466, A325353, A325354, A325358, A325391, A325396, A325399, A325404, A325406, A325457, A325468.

%K nonn

%O 0,4

%A _Gus Wiseman_, May 02 2019