Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #9 Aug 06 2023 08:17:52
%S 1,1,2,3,4,4,8,7,11,13,17,18,32,30,44,54,70,78,114,125,171,205,257,
%T 302,408,464,592,711,892,1042,1330,1543,1925,2279,2787,3291,4061,4727,
%U 5753,6792,8197,9583,11593,13505,16198,18965,22548,26290,31340,36363,43046
%N Number of integer partitions of n containing all of their own nonzero first differences.
%e The partition (10,5,3,3,2,1) has nonzero differences (5,2,1,1) so is counted under a(24).
%e The a(1) = 1 through a(9) = 13 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (11) (21) (22) (221) (33) (421) (44) (63)
%e (111) (211) (2111) (42) (2221) (422) (333)
%e (1111) (11111) (222) (3211) (2222) (3321)
%e (321) (22111) (3221) (4221)
%e (2211) (211111) (4211) (22221)
%e (21111) (1111111) (22211) (32211)
%e (111111) (32111) (42111)
%e (221111) (222111)
%e (2111111) (321111)
%e (11111111) (2211111)
%e (21111111)
%e (111111111)
%t Table[Length[Select[IntegerPartitions[n], SubsetQ[#,Differences[Union[#]]]&]],{n,0,30}]
%Y For no differences we have A363260, subsets A364463, strict A364464.
%Y For at least one difference we have A364467, ranks A364537, strict A364536.
%Y For subsets instead of partitions we have A364671, complement A364672.
%Y The strict case (no differences of 0) is counted by A364673.
%Y For submultisets instead of subsets we have A364675.
%Y A000041 counts integer partitions, strict A000009.
%Y A008284 counts partitions by length, strict A008289.
%Y A236912 counts sum-free partitions w/o re-used parts, complement A237113.
%Y A325325 counts partitions with distinct first differences.
%Y Cf. A002865, A007862, A025065, A229816, A237667, A320347, A326083, A363225, A364272, A364466.
%K nonn
%O 0,3
%A _Gus Wiseman_, Aug 04 2023