OFFSET
0,3
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 A325394.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1500
EXAMPLE
The a(1) = 1 through a(8) = 6 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (32) (33) (43) (44)
(1111) (11111) (222) (1111111) (53)
(111111) (332)
(2222)
(11111111)
For example, the augmented differences of (6,6,5,3) are (1,2,3,3), which are weakly increasing, so (6,6,5,3) is counted under a(20).
MATHEMATICA
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], OrderedQ[aug[#]]&]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 23 2019
STATUS
approved