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A325358 Number of integer partitions of n whose augmented differences are strictly decreasing.

%I A325358 #8 Mar 04 2021 03:18:38
%S A325358 1,1,1,2,2,2,3,4,4,5,6,6,7,9,10,11,13,14,15,18,20,21,24,26,28,33,36,
%T A325358 38,43,46,49,56,60,63,71,76,80,90,96,100,112,120,125,139,149,155,171,
%U A325358 183,190,208,223,232,252,269,280,304,325,338,364,387,403
%N A325358 Number of integer partitions of n whose augmented differences are strictly decreasing.
%C A325358 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 A325358 The Heinz numbers of these partitions are given by A325396.
%H A325358 Fausto A. C. Cariboni, <a href="/A325358/b325358.txt">Table of n, a(n) for n = 0..1000</a>
%H A325358 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325358 The a(1) = 1 through a(11) = 6 partitions:
%e A325358   (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (10)   (11)
%e A325358             (21)  (31)  (41)  (42)  (52)   (62)   (63)   (73)   (83)
%e A325358                               (51)  (61)   (71)   (72)   (82)   (92)
%e A325358                                     (421)  (521)  (81)   (91)   (101)
%e A325358                                                   (621)  (631)  (731)
%e A325358                                                          (721)  (821)
%t A325358 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A325358 Table[Length[Select[IntegerPartitions[n],OrderedQ[aug[#],Greater]&]],{n,0,30}]
%Y A325358 Cf. A049988, A240026, A320466, A325349, A325350, A325351, A325356, A325357, A325359, A325393.
%K A325358 nonn
%O A325358 0,4
%A A325358 _Gus Wiseman_, May 01 2019