A325769 - OEIS

%I #14 Feb 25 2021 02:27:04

%S 1,1,2,3,4,6,7,11,12,17,19,29,28,41,42,62,61,88,87,123,121,168,164,

%T 234,225,306,306,411,401,527,533,700,689,894,885,1163,1150,1452,1469,

%U 1866,1835,2333,2346,2913,2913,3638,3619,4511,4537,5497,5576,6859,6827,8263

%N Number of integer partitions of n whose distinct consecutive subsequences have different sums.

%C For example (3,3,1,1) is counted under a(8) because it has distinct consecutive subsequences (), (1), (1,1), (3), (3,1), (3,1,1), (3,3), (3,3,1), (3,3,1,1), all of which have different sums.

%C The Heinz numbers of these partitions are given by A325778.

%H Fausto A. C. Cariboni, <a href="/A325769/b325769.txt">Table of n, a(n) for n = 0..300</a>

%e The a(1) = 1 through a(8) = 12 partitions:

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

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

%e              (111)  (31)    (41)     (42)      (52)       (53)

%e                     (1111)  (221)    (51)      (61)       (62)

%e                             (311)    (222)     (322)      (71)

%e                             (11111)  (411)     (331)      (332)

%e                                      (111111)  (421)      (521)

%e                                                (511)      (611)

%e                                                (2221)     (2222)

%e                                                (4111)     (3311)

%e                                                (1111111)  (5111)

%e                                                           (11111111)

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@Total/@Union[ReplaceList[#,{___,s__,___}:>{s}]]&]],{n,0,30}]

%Y Cf. A000041, A002033, A143823, A169942, A325676, A325677, A325683, A325768, A325770, A325778.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 21 2019

%E a(41)-a(53) from _Fausto A. C. Cariboni_, Feb 24 2021