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Number of integer partitions of n whose augmented differences are strictly decreasing.
10

%I #8 Mar 04 2021 03:18:38

%S 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 38,43,46,49,56,60,63,71,76,80,90,96,100,112,120,125,139,149,155,171,

%U 183,190,208,223,232,252,269,280,304,325,338,364,387,403

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

%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 A325396.

%H Fausto A. C. Cariboni, <a href="/A325358/b325358.txt">Table of n, a(n) for n = 0..1000</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) = 6 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

%e (21) (31) (41) (42) (52) (62) (63) (73) (83)

%e (51) (61) (71) (72) (82) (92)

%e (421) (521) (81) (91) (101)

%e (621) (631) (731)

%e (721) (821)

%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[#],Greater]&]],{n,0,30}]

%Y Cf. A049988, A240026, A320466, A325349, A325350, A325351, A325356, A325357, A325359, A325393.

%K nonn

%O 0,4

%A _Gus Wiseman_, May 01 2019