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A006363
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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)
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0
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1, 168, 7581, 160948, 2068224, 18561984, 127234008, 706987164, 3320153661, 13583619496, 49530070161, 163806121656, 498180781144, 1408758106368, 3737505070344, 9372218674824, 22351423903953, 50960797533096, 111574385244253, 235475590500876, 480631725411720, 951504952784320, 1831615165328400, 3435931869872580
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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