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A325356
Number of integer partitions of n whose augmented differences are weakly increasing.
14
1, 1, 2, 2, 3, 3, 4, 3, 6, 5, 5, 6, 8, 6, 10, 9, 8, 10, 13, 10, 15, 14, 13, 15, 21, 15, 19, 21, 20, 25, 25, 20, 31, 30, 30, 32, 35, 28, 40, 44, 36, 42, 50, 43, 54, 53, 49, 57, 67, 58, 68, 66, 66, 78, 84, 71, 86, 92, 82, 99, 109
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 23 2019
STATUS
approved