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.

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

Offset: 0

Author

Henry Bottomley, Feb 17 2002

Keywords

• nonn
• tabl

Comments

The number of partitions of m into up to n positive integers each no more than k is maximized for given n and k by m=floor(nk/2) or ceiling(nk/2) (and possibly some other values).

Examples

			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.
		

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.

Programs

Maple

A067059 := proc(n,k)
    local m,a1,a2 ;
    a1 := 0 ;
    m := floor(n*k/2) ;
    for L in combinat[partition](m) do
        if nops(L) <= n then
            if max(op(L)) <= k then
                a1 := a1+1 ;
            end if ;
        end if;
    end do:
    a2 := 0 ;
    m := ceil(n*k/2) ;
    for L in combinat[partition](m) do
        if nops(L) <= n then
            if max(op(L)) <= k then
                a2 := a2+1 ;
            end if ;
        end if;
    end do:
    max(a1,a2) ;
end proc:
for d from 0 to 12 do
    for k from 0 to d do
        printf("%d,",A067059(d-k,k)) ;
    end do:
end do: # R. J. Mathar, Nov 13 2016

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}]] (* Jean-François Alcover, Jan 02 2012 *)

Sage

def A067059(n, k):
    return Partitions((n*k)//2, max_length=n, max_part=k).cardinality()
for n in (0..9): [A067059(n,k) for k in (0..9)] # Peter Luschny, May 05 2014