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%I #10 Feb 28 2021 12:09:47
%S 1,1,1,2,3,2,4,5,7,7,12,10,13,15,21,21,31,34,38,45,55,60,71,80,84,103,
%T 119,134,152,186,192,228,263,292,321,377,399,454,514,565,618,709,752,
%U 840,958,1050,1140,1297,1402,1568,1755,1901,2080,2343,2524,2758,3074
%N Number of integer partitions of n whose augmented differences are distinct.
%C The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
%C The Heinz numbers of these partitions are given by A325366.
%H Fausto A. C. Cariboni, <a href="/A325349/b325349.txt">Table of n, a(n) for n = 0..440</a>
%e The a(1) = 1 through a(11) = 10 partitions (A = 10, B = 11):
%e (1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)
%e (21) (22) (41) (33) (43) (44) (54) (55) (65)
%e (31) (42) (52) (62) (63) (64) (83)
%e (51) (61) (71) (72) (73) (92)
%e (421) (422) (81) (82) (A1)
%e (431) (522) (91) (443)
%e (521) (621) (433) (641)
%e (442) (722)
%e (541) (731)
%e (622) (821)
%e (631)
%e (721)
%e For example, (4,4,3) has augmented differences (1,2,3), which are distinct, so (4,4,3) is counted under a(11).
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@Differences[Append[#,1]]&]],{n,0,30}]
%Y Cf. A000837, A049988, A098859, A325324, A325325, A325328, A325351, A325366, A325404.
%K nonn
%O 0,4
%A _Gus Wiseman_, Apr 23 2019
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