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Number of integer partitions of n with distinct differences between successive parts.
132

%I #19 Feb 27 2021 22:04:28

%S 1,1,2,2,4,5,5,8,11,12,16,22,21,30,34,42,49,64,67,87,95,117,132,160,

%T 169,207,230,274,301,360,395,463,506,602,656,762,834,960,1042,1220,

%U 1311,1505,1643,1859,2000,2341,2491,2827,3083,3464,3747,4302,4561,5154

%N Number of integer partitions of n with distinct differences between successive parts.

%C The Heinz numbers of these partitions are given by A325368.

%H Fausto A. C. Cariboni, <a href="/A325325/b325325.txt">Table of n, a(n) for n = 0..400</a> (terms 0..123 from Alois P. Heinz)

%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(0) = 1 through a(9) = 12 partitions:

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

%e (11) (21) (22) (32) (33) (43) (44) (54)

%e (31) (41) (42) (52) (53) (63)

%e (211) (221) (51) (61) (62) (72)

%e (311) (411) (322) (71) (81)

%e (331) (332) (441)

%e (421) (422) (522)

%e (511) (431) (621)

%e (521) (711)

%e (611) (4221)

%e (4211) (4311)

%e (5211)

%e For example, (5,2,1,1) has differences (-3,-1,0), which are distinct, so (5,2,1,1) is counted under a(9).

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

%Y Cf. A007294, A007862, A049988, A098859, A240026, A240027, A320348, A320466, A320470, A325324, A325349, A325352, A325368, A325404, A325468.

%K nonn

%O 0,3

%A _Gus Wiseman_, Apr 23 2019