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A157046
Maximum number of partitions of n into exactly k parts, each <= k. a(n) is maximum in each row of A157044.
3
1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 16, 19, 24, 31, 37, 46, 58, 70, 86, 104, 127, 156, 185, 222, 273, 326, 392, 463, 556, 669, 792, 939, 1109, 1317, 1564, 1838, 2156, 2535, 2986, 3514, 4100, 4777, 5577, 6526, 7621, 8847, 10251, 11869, 13807, 16032, 18529, 21370
OFFSET
0,6
COMMENTS
Without the constraint on each part being <= k: see A008284 and A002569.
LINKS
EXAMPLE
For n=9 the counts of partitions for k=1..9 is 0,0,1,4,5,3,2,1,1 so the maximum is 5 (at k=5).
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, 1,
`if`(i*t<n, 0, b(n, i-1, t)+b(n-i, min(n-i, i), t-1)))
end:
a:= n-> max(seq(b(n-i, min(n-i, i-1), i), i=0..n)):
seq(a(n), n=0..55); # Alois P. Heinz, May 12 2022
MATHEMATICA
Max @@@ Table[T[n, k, k]-T[n, k-1, k], {n, 1, 128}, {k, n}] (* with T[n, a, b] as defined in A047993 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Wouter Meeussen, Feb 22 2009
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, May 12 2022
STATUS
approved