0,3

Par(n) is the set of partitions of n under 'dominance order': partition P is <= partition Q iff the sum of the largest k parts of P is <= the corresponding sum for Q for all k.

For n=10, the maximum size is A076269(10)=4. There are 2 maximum-size antichains: {5+1+1+1+1+1, 4+3+1+1+1, 4+2+2+2, 3+3+3+1} and {6+1+1+1+1, 5+2+2+1, 4+4+1+1, 4+3+3}. So a(10)=2.

leq[p_, q_] := If[Length[p]<Length[q], False, Module[{i, ds}, For[i=1; ds=0, i<Length[q], i++, If[(ds+=q[[i]]-p[[i]])<0, Return[False]]]; True]]; maxac[l_] := If[Length[l]<=1, Length[l], maxac[l]=Max[maxac[Drop[l, 1]], 1+maxac[Select[l, !leq[ #, l[[1]]]&&!leq[l[[1]], # ]&]]]]; maxantichains[l_] := If[Length[l]<=1, {l}, Module[{v, t}, v={}; If[maxac[l]==maxac[Drop[l, 1]], v=Join[v, maxantichains[Drop[l, 1]]]]; t=Select[l, !leq[ #, l[[1]]]&&!leq[l[[1]], # ]&]; If[maxac[l]==1+maxac[t], v=Join[v, Prepend[ #, l[[1]]]&/@maxantichains[t]]]; v]]; a[n_] := Length[maxantichains[Partitions[n]]] (* First do <<DiscreteMath`Combinatorica` *) (* maxac[l] = size of largest antichain in set l. maxantichains[l] = list of all maximum-length antichains in l. *)

The corresponding sizes are in A076269.

Adjacent sequences: A077762 A077763 A077764 this_sequence A077766 A077767 A077768

Sequence in context: A038194 A111309 A007605 this_sequence A078400 A068951 A139752

nonn,more

Dean Hickerson (dean(AT)math.ucdavis.edu), Nov 14 2002