OFFSET
0,7
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
T. Brylawski, The lattice of integer partitions, Discrete Math. 6 (1973), 201-219.
Edward Early, Chain Lengths in the Dominance Lattice, June 8, 2013.
Edward Early, Chain Lengths in the Dominance Lattice, Discrete Mathematics, Volume 313, Issue 20, 28 October 2013, Pages 2168-2177.
C. Greene and D. J. Kleitman, Longest Chains in the Lattice of Integer Partitions ordered by Majorization, Europ. J. Combinatorics 7 (1986), 1-10.
Grant Kopitzke, The Gini Index of an Integer Partition, arXiv:2005.04284 [math.CO], 2020. Mentions this sequence.
FORMULA
Order of growth is between n^(-5/2)e^(Pi*sqrt(2n/3)) and n^(-1)e^(Pi*sqrt(2n/3)).
EXAMPLE
a(10)=4; one antichain consists of 5+1+1+1+1+1, 4+3+1+1+1, 4+2+2+2 and 3+3+3+1.
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]], # ]&]]]]; a[n_] := a[n]=maxac[IntegerPartitions[n]]
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Edward Early, Nov 05 2002
EXTENSIONS
Edited by Dean Hickerson, Nov 09 2002
a(22)-a(26) by Paul Tabatabai, Dec 05 2018
STATUS
approved