|
| |
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
|
|
|
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}]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
