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%I #9 Mar 03 2021 19:28:59
%S 1,1,2,2,3,3,4,3,6,5,5,6,8,6,10,9,8,10,13,10,15,14,13,15,21,15,19,21,
%T 20,25,25,20,31,30,30,32,35,28,40,44,36,42,50,43,54,53,49,57,67,58,68,
%U 66,66,78,84,71,86,92,82,99,109
%N Number of integer partitions of n whose augmented differences are weakly increasing.
%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 A325394.
%H Fausto A. C. Cariboni, <a href="/A325356/b325356.txt">Table of n, a(n) for n = 0..1500</a>
%e The a(1) = 1 through a(8) = 6 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (111) (22) (32) (33) (43) (44)
%e (1111) (11111) (222) (1111111) (53)
%e (111111) (332)
%e (2222)
%e (11111111)
%e For example, the augmented differences of (6,6,5,3) are (1,2,3,3), which are weakly increasing, so (6,6,5,3) is counted under a(20).
%t aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t Table[Length[Select[IntegerPartitions[n],OrderedQ[aug[#]]&]],{n,0,30}]
%Y Cf. A000837, A007294, A049988, A098859, A325350, A325351, A325354, A325357, A325358, A325360, A325394.
%K nonn
%O 0,3
%A _Gus Wiseman_, Apr 23 2019
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