|
|
A183558
|
|
Number of partitions of n containing a clique of size 1.
|
|
29
|
|
|
0, 1, 1, 2, 3, 6, 7, 13, 16, 25, 33, 49, 61, 90, 113, 156, 198, 269, 334, 448, 556, 726, 902, 1163, 1428, 1827, 2237, 2817, 3443, 4302, 5219, 6478, 7833, 9632, 11616, 14197, 17031, 20712, 24769, 29925, 35688, 42920, 50980, 61059, 72318, 86206, 101837, 120941
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: (1-Product_{j>0} (1-x^(j)+x^(2*j))) / (Product_{j>0} (1-x^j)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). (End)
|
|
EXAMPLE
|
a(5) = 6, because 6 partitions of 5 contain (at least) one clique of size 1: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
The a(1) = 1 through a(8) = 16 partitions are the following. The Heinz numbers of these partitions are given by A052485 (weak numbers).
(1) (2) (3) (4) (5) (6) (7) (8)
(21) (31) (32) (42) (43) (53)
(211) (41) (51) (52) (62)
(221) (321) (61) (71)
(311) (411) (322) (332)
(2111) (3111) (331) (422)
(21111) (421) (431)
(511) (521)
(2221) (611)
(3211) (3221)
(4111) (4211)
(31111) (5111)
(211111) (32111)
(41111)
(311111)
(2111111)
(End)
|
|
MAPLE
|
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->`if`(j=1, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);
|
|
MATHEMATICA
|
max = 50; f = (1 - Product[1 - x^j + x^(2*j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; CoefficientList[s, x] (* Jean-François Alcover, Oct 01 2014. Edited by Gus Wiseman, Apr 19 2019 *)
|
|
CROSSREFS
|
Cf. A000041, A007690, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|