A006363 Number of antichains (or order ideals) in the poset B_4 X [n]; or size of the distributive lattice J(B_4 X [n]).

1, 168, 7581, 160948, 2068224, 18561984, 127234008, 706987164, 3320153661, 13583619496, 49530070161, 163806121656, 498180781144, 1408758106368, 3737505070344, 9372218674824, 22351423903953, 50960797533096, 111574385244253, 235475590500876, 480631725411720, 951504952784320, 1831615165328400, 3435931869872580

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) is the number of order preserving maps from B_4 into [n+1]. a(n) is also the number of length n+1 multichains from bottom to top in J(B_4). See Stanley reference for bijections with description in title. - Geoffrey Critzer, Jan 15 2021

References

• J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, page 256, Proposition 3.5.1.

Links

• J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
• G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.

Crossrefs

Cf. A056932, A002415.

Programs

Mathematica

p = Subsets[Range[4]];
f[list1_, list2_] := If[ContainsAll[list2, list1], 1, 0]; \[Zeta] = Table[Table[f[p[[i]], p[[j]]], {j, 1, 16}], {i, 1, 16}]; JB4 =
Complement[Subsets[Range[16]],Level[Table[Select[Subsets[Range[16]],MemberQ[#, i] && !ContainsAll[Level[Position[\[Zeta][[All, i]], 1], {2}]][#] &], {i, 2,16}], {2}] // DeleteDuplicates]; \[Zeta]JB4 =Table[Table[f[JB4[[i]], JB4[[j]]], {j, 1, 168}], {i, 1,168}]; \[CapitalOmega][n_] := Expand[InterpolatingPolynomial[
Table[{k, MatrixPower[\[Zeta]JB4, k][[1, 168]]}, {k, 1, 17}],n]]; Table[\[CapitalOmega][n], {n, 1, 30}] (* Geoffrey Critzer, Jan 15 2021 *)

Extensions

Title corrected by Geoffrey Critzer, Jan 15 2021

a(11)-a(23) from Geoffrey Critzer, Jan 15 2021