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A325357
Number of integer partitions of n whose augmented differences are strictly increasing.
11
1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 3, 5, 5, 4, 5, 6, 5, 7, 7, 7, 7, 9, 7, 10, 10, 8, 11, 13, 10, 13, 14, 12, 14, 17, 13, 17, 19, 17, 18, 22, 19, 22, 24, 21, 24, 28, 24, 29, 30, 28, 31, 35, 30, 35, 40, 36
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 23 2019
STATUS
approved