A076269 Size of largest antichain in partition lattice Par(n).

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 7, 9, 10, 11, 14, 17, 20, 24, 29, 35, 40, 48, 55

Offset: 0

Author

Edward Early, Nov 05 2002

Keywords

• nonn
• hard
• more

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.

Examples

			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.
		

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.

Crossrefs

Cf. A006463, A077765, A076779.

Programs

Mathematica

leq[p_, q_] := If[Length[p]
				

Formula

Order of growth is between n^(-5/2)e^(Pi*sqrt(2n/3)) and n^(-1)e^(Pi*sqrt(2n/3)).

Extensions

Edited by Dean Hickerson, Nov 09 2002

a(22)-a(26) by Paul Tabatabai, Dec 05 2018