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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 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-François 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 October 22 17:55 EDT 2019. Contains 328319 sequences. (Running on oeis4.)