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A077765 Number of maximum-size antichains in partition lattice Par(n). 2
1, 1, 2, 3, 5, 7, 2, 4, 15, 4, 2, 11, 18, 14, 53, 2, 54, 1606, 482, 104, 754, 536 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..21.

EXAMPLE

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.

MATHEMATICA

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. *)

CROSSREFS

The corresponding sizes are in A076269.

Sequence in context: A038194 A111309 A007605 * A078400 A068951 A139752

Adjacent sequences:  A077762 A077763 A077764 * A077766 A077767 A077768

KEYWORD

nonn,more

AUTHOR

Dean Hickerson, Nov 14 2002

STATUS

approved

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Last modified January 19 16:16 EST 2021. Contains 340270 sequences. (Running on oeis4.)