%I #19 Feb 27 2021 12:06:47

%S 1,1,2,1,3,4,4,7,7,7,10,15,13,22,25,26,31,43,39,55,54,68,75,98,97,128,

%T 135,165,177,217,223,277,282,339,356,438,444,527,553,667,694,816,868,

%U 1015,1054,1279,1304,1538,1631,1849,1958,2304,2360,2701,2899,3267

%N Number of integer partitions of n whose differences (with the last part taken to be 0) are distinct.

%C The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).

%C The Heinz numbers of these partitions are given by A325367.

%H Fausto A. C. Cariboni, <a href="/A325324/b325324.txt">Table of n, a(n) for n = 0..400</a>

%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(11) = 15 partitions (A = 10, B = 11):

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

%e (11) (22) (32) (33) (43) (44) (54) (55) (65)

%e (31) (41) (51) (52) (53) (72) (64) (74)

%e (311) (411) (61) (62) (81) (73) (83)

%e (322) (71) (441) (82) (92)

%e (331) (332) (522) (91) (A1)

%e (511) (611) (711) (433) (443)

%e (622) (533)

%e (631) (551)

%e (811) (632)

%e (641)

%e (722)

%e (731)

%e (911)

%e (6311)

%e For example, (6,3,1,1) has differences (-3,-2,0,-1), which are distinct, so (6,3,1,1) is counted under a(11).

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@Differences[Append[#,0]]&]],{n,0,30}]

%Y Cf. A007862, A049988, A098859, A130091, A240026, A320348, A320466, A320509, A325325, A325349, A325366, A325367, A325368, A325404, A325407.

%K nonn

%O 0,3

%A _Gus Wiseman_, Apr 23 2019