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Number of integer partitions of n containing all of their own nonzero first differences.
5

%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