A056932 Antichains (or order ideals) in the poset 2*2*2*n or size of the distributive lattice J(2*2*2*n).

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

Offset: 0

Author

Mitch Harris

Keywords

• nonn
• easy

Comments

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

References

• 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.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• 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.
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 9.
• Index entries for sequences related to posets
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A000372, A006360, A006361, A006362, A056933, A056934, A056935, A056936, A056937.

Programs

Mathematica

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 *)

Formula

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]