|
|
A342498
|
|
Number of integer partitions of n with strictly increasing first quotients.
|
|
5
|
|
|
1, 1, 2, 2, 4, 4, 5, 6, 8, 9, 12, 12, 14, 16, 18, 20, 24, 26, 27, 30, 35, 37, 45, 47, 52, 56, 61, 65, 72, 77, 83, 90, 95, 99, 109, 117, 127, 135, 144, 151, 164, 172, 181, 197, 209, 222, 239, 249, 263, 280, 297, 310, 332, 349, 368, 391, 412, 433, 457, 480, 503
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Also the number of reversed integer partitions of n with strictly increasing first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).
|
|
LINKS
|
|
|
EXAMPLE
|
The partition y = (13,7,2,1) has first quotients (7/13,2/7,1/2) so is not counted under a(23). However, the first differences (-6,-5,-1) are strictly increasing, so y is counted under A240027(23).
The a(1) = 1 through a(9) = 9 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(11) (21) (22) (32) (33) (43) (44) (54)
(31) (41) (42) (52) (53) (63)
(211) (311) (51) (61) (62) (72)
(411) (322) (71) (81)
(511) (422) (522)
(521) (621)
(611) (711)
(5211)
|
|
MATHEMATICA
|
Table[Length[Select[IntegerPartitions[n], Less@@Divide@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 30}]
|
|
CROSSREFS
|
The version for differences instead of quotients is A240027.
The weakly increasing version is A342497.
The strictly decreasing version is A342499.
The Heinz numbers of these partitions are A342524.
A000005 counts constant partitions.
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342098 counts partitions with adjacent parts x > 2y.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|