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A056932
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Antichains (or order ideals) in the poset 2*2*2*n or size of the distributive lattice J(2*2*2*n).
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14
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1, 20, 168, 887, 3490, 11196, 30900, 75966, 170379, 354640, 693836, 1288365, 2287844, 3908776, 6456600, 10352796, 16167765, 24660252, 36824128, 53943395, 77656326, 110029700, 153644140, 211691610, 288086175, 387589176, 515950020, 680063833, 888147272
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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_3 into [n+1]. a(n) is also the number of length n+1 multichains from bottom to top in J(B_3). See Stanley reference for bijections with description in title. - Geoffrey Critzer, Jan 07 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.
Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.
R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, page 256, Proposition 3.5.1.
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LINKS
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FORMULA
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a(n) = 48*C(n+8, 8) - 96*C(n+7, 7) + 63*C(n+6, 6) - 15*C(n+5, 5) + C(n+4, 4).
G.f.: (1+11*x+24*x^2+11*x^3+x^4)/(1-x)^9. [Berman and Koehler]
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MATHEMATICA
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Table[48*Binomial[n+8, 8] - 96*Binomial[n+7, 7] + 63*Binomial[n+6, 6] - 15*Binomial[n+5, 5] + Binomial[n+4, 4], {n, 0, nn}] (* T. D. Noe, May 29 2012 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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