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A325349
Number of integer partitions of n whose augmented differences are distinct.
16
1, 1, 1, 2, 3, 2, 4, 5, 7, 7, 12, 10, 13, 15, 21, 21, 31, 34, 38, 45, 55, 60, 71, 80, 84, 103, 119, 134, 152, 186, 192, 228, 263, 292, 321, 377, 399, 454, 514, 565, 618, 709, 752, 840, 958, 1050, 1140, 1297, 1402, 1568, 1755, 1901, 2080, 2343, 2524, 2758, 3074
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 23 2019
STATUS
approved