|
| |
|
|
A067059
|
|
Square array read by antidiagonals of partitions which half fill an n*k box, i.e. partitions of floor[nk/2] or ceiling[nk/2] into up to n positive integers each no more than k.
|
|
12
|
|
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 1, 4, 6, 8, 6, 4, 1, 1, 1, 1, 4, 8, 12, 12, 8, 4, 1, 1, 1, 1, 5, 10, 18, 20, 18, 10, 5, 1, 1, 1, 1, 5, 13, 24, 32, 32, 24, 13, 5, 1, 1, 1, 1, 6, 15, 33, 49, 58, 49, 33, 15, 6, 1, 1, 1, 1, 6
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,13
|
|
|
COMMENTS
|
The number of partitions of m into up to n positive integers each no more than k is maximised for given n and k by m=floor[nk/2] or ceiling[nk/2] (and possibly some other values).
|
|
|
LINKS
|
Table of n, a(n) for n=0..93.
|
|
|
EXAMPLE
|
Rows start:
1,1,1,1,1,1,...;
1,1,1,1,1,1,...;
1,1,2,2,3,3,...;
1,1,2,3,5,6,...;
1,1,3,5,8,12,...; etc.
T(4,5)=12 since 10 can be partitioned into
5+5, 5+4+1, 5+3+2, 5+3+1+1, 5+2+2+1, 4+4+2, 4+3+3,
4+4+1+1, 4+3+2+1, 4+2+2+2, 3+3+3+1, and 3+3+2+2.
|
|
|
MATHEMATICA
|
t[n_, k_] := Length[ IntegerPartitions[ Floor[n*k/2], n, Range[k]]]; Flatten[ Table[ t[n-k , k], {n, 0, 13}, {k, 0, n}]] (* From Jean-François Alcover, Jan 02 2012 *)
|
|
|
CROSSREFS
|
As this is symmetric, rows and columns each include A000012 twice, A008619, A001971, A001973, A001975, A001977, A001979 and A001981. Diagonal is A029895. T(n, n*(n-1)) is the magic series A052456.
Sequence in context: A057790 A224697 A052307 * A049704 A047996 A063686
Adjacent sequences: A067056 A067057 A067058 * A067060 A067061 A067062
|
|
|
KEYWORD
|
nonn,tabl
|
|
|
AUTHOR
|
Henry Bottomley, Feb 17 2002
|
|
|
STATUS
|
approved
|
| |
|
|
