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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. 14
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 maximized 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.

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-Fran├žois Alcover, Jan 02 2012 *)

PROG

(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

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 A227690

Adjacent sequences:  A067056 A067057 A067058 * A067060 A067061 A067062

KEYWORD

nonn,tabl

AUTHOR

Henry Bottomley, Feb 17 2002

STATUS

approved

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Last modified July 23 08:59 EDT 2017. Contains 289686 sequences.