OFFSET
0,4
COMMENTS
The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
The Heinz numbers of these partitions are given by A325366.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..440
EXAMPLE
The a(1) = 1 through a(11) = 10 partitions (A = 10, B = 11):
(1) (2) (3) (4) (5) (6) (7) (8) (9) (A) (B)
(21) (22) (41) (33) (43) (44) (54) (55) (65)
(31) (42) (52) (62) (63) (64) (83)
(51) (61) (71) (72) (73) (92)
(421) (422) (81) (82) (A1)
(431) (522) (91) (443)
(521) (621) (433) (641)
(442) (722)
(541) (731)
(622) (821)
(631)
(721)
For example, (4,4,3) has augmented differences (1,2,3), which are distinct, so (4,4,3) is counted under a(11).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@Differences[Append[#, 1]]&]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 23 2019
STATUS
approved