cp's OEIS Frontend

A364674 Number of integer partitions of n containing all of their own nonzero first differences.

%I A364674 #9 Aug 06 2023 08:17:52
%S A364674 1,1,2,3,4,4,8,7,11,13,17,18,32,30,44,54,70,78,114,125,171,205,257,
%T A364674 302,408,464,592,711,892,1042,1330,1543,1925,2279,2787,3291,4061,4727,
%U A364674 5753,6792,8197,9583,11593,13505,16198,18965,22548,26290,31340,36363,43046
%N A364674 Number of integer partitions of n containing all of their own nonzero first differences.
%e A364674 The partition (10,5,3,3,2,1) has nonzero differences (5,2,1,1) so is counted under a(24).
%e A364674 The a(1) = 1 through a(9) = 13 partitions:
%e A364674   (1) (2)  (3)   (4)    (5)     (6)      (7)       (8)        (9)
%e A364674       (11) (21)  (22)   (221)   (33)     (421)     (44)       (63)
%e A364674            (111) (211)  (2111)  (42)     (2221)    (422)      (333)
%e A364674                  (1111) (11111) (222)    (3211)    (2222)     (3321)
%e A364674                                 (321)    (22111)   (3221)     (4221)
%e A364674                                 (2211)   (211111)  (4211)     (22221)
%e A364674                                 (21111)  (1111111) (22211)    (32211)
%e A364674                                 (111111)           (32111)    (42111)
%e A364674                                                    (221111)   (222111)
%e A364674                                                    (2111111)  (321111)
%e A364674                                                    (11111111) (2211111)
%e A364674                                                               (21111111)
%e A364674                                                               (111111111)
%t A364674 Table[Length[Select[IntegerPartitions[n], SubsetQ[#,Differences[Union[#]]]&]],{n,0,30}]
%Y A364674 For no differences we have A363260, subsets A364463, strict A364464.
%Y A364674 For at least one difference we have A364467, ranks A364537, strict A364536.
%Y A364674 For subsets instead of partitions we have A364671, complement A364672.
%Y A364674 The strict case (no differences of 0) is counted by A364673.
%Y A364674 For submultisets instead of subsets we have A364675.
%Y A364674 A000041 counts integer partitions, strict A000009.
%Y A364674 A008284 counts partitions by length, strict A008289.
%Y A364674 A236912 counts sum-free partitions w/o re-used parts, complement A237113.
%Y A364674 A325325 counts partitions with distinct first differences.
%Y A364674 Cf. A002865, A007862, A025065, A229816, A237667, A320347, A326083, A363225, A364272, A364466.
%K A364674 nonn
%O A364674 0,3
%A A364674 _Gus Wiseman_, Aug 04 2023