A157046 Maximum number of partitions of n into exactly k parts, each <= k. a(n) is maximum in each row of A157044.

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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

Cf. A157044, A047993, A008284, A002569.

Sequence in context: A258875 A179241 A316082 * A017979 A140881 A316081

Adjacent sequences: A157043 A157044 A157045 * A157047 A157048 A157049

Keyword

nonn

Author

Wouter Meeussen, Feb 22 2009

Extensions

a(0)=1 prepended by Alois P. Heinz, May 12 2022

Status

approved