OFFSET
0,5
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 A325395.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000
EXAMPLE
The a(28) = 10 partitions:
(28)
(18,10)
(17,11)
(16,12)
(15,13)
(14,14)
(12,10,6)
(11,10,7)
(10,10,8)
(8,8,7,5)
For example, the augmented differences of (8,8,7,5) are (1,2,3,5), which are strictly increasing.
MATHEMATICA
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], Less@@aug[#]&]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 23 2019
STATUS
approved