|
| |
|
|
A183567
|
|
Number of partitions of n containing a clique of size 10.
|
|
12
|
|
|
|
1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 13, 15, 22, 26, 37, 45, 61, 74, 99, 120, 157, 192, 247, 299, 381, 462, 580, 703, 874, 1055, 1303, 1569, 1921, 2309, 2808, 3363, 4070, 4859, 5848, 6964, 8342, 9903, 11817, 13988, 16623, 19626, 23240, 27363, 32297
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
10,5
|
|
|
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^(10*j)+x^(11*j))) / (Product_{j>0} (1-x^j)).
|
|
|
EXAMPLE
|
a(14) = 2, because 2 partitions of 14 contain (at least) one clique of size 10: [1,1,1,1,1,1,1,1,1,1,2,2], [1,1,1,1,1,1,1,1,1,1,4].
|
|
|
MAPLE
|
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->`if`(j=10, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=10..60);
|
|
|
MATHEMATICA
|
max = 60; f = (1 - Product[1 - x^(10j) + x^(11j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 10] (* Jean-François Alcover, Oct 01 2014 *)
Table[Length[Select[IntegerPartitions[n], MemberQ[Length/@Split[#], 10]&]], {n, 10, 60}] (* Harvey P. Dale, Oct 02 2021 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
