A325769 Number of integer partitions of n whose distinct consecutive subsequences have different sums.

1, 1, 2, 3, 4, 6, 7, 11, 12, 17, 19, 29, 28, 41, 42, 62, 61, 88, 87, 123, 121, 168, 164, 234, 225, 306, 306, 411, 401, 527, 533, 700, 689, 894, 885, 1163, 1150, 1452, 1469, 1866, 1835, 2333, 2346, 2913, 2913, 3638, 3619, 4511, 4537, 5497, 5576, 6859, 6827, 8263

Offset

0,3

Comments

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.

The Heinz numbers of these partitions are given by A325778.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300

Example

The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (311)    (222)     (322)      (71)
                            (11111)  (411)     (331)      (332)
                                     (111111)  (421)      (521)
                                               (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (1111111)  (5111)
                                                          (11111111)

Mathematica

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

Crossrefs

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

Sequence in context: A203898 A108917 A237667 * A360955 A226538 A339510

Adjacent sequences: A325766 A325767 A325768 * A325770 A325771 A325772

Keyword

nonn

Author

Gus Wiseman, May 21 2019

Extensions

a(41)-a(53) from Fausto A. C. Cariboni, Feb 24 2021

Status

approved