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]). (Formerly M5408)

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

0,2

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

Table of n, a(n) for n=0..23.

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.

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 *)

Crossrefs

Cf. A056932, A002415.

Sequence in context: A003800 A373501 A278010 * A271033 A019286 A218413

Adjacent sequences: A006360 A006361 A006362 * A006364 A006365 A006366

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Title corrected by Geoffrey Critzer, Jan 15 2021

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

Status

approved