A056005 Number of 3-element ordered antichains on an unlabeled n-element set; T_1-hypergraphs with 3 labeled nodes and n hyperedges.
0, 0, 0, 2, 19, 90, 302, 820, 1926, 4068, 7920, 14454, 25025, 41470, 66222, 102440, 154156, 226440, 325584, 459306, 636975, 869858, 1171390, 1557468, 2046770, 2661100, 3425760, 4369950, 5527197, 6935814, 8639390, 10687312, 13135320, 16046096, 19489888
Offset: 0
Examples
There are 19 3-element ordered antichains on an unlabeled 4-element set: ({4},{3},{2}), ({4},{3},{1,2}), ({4},{2,3},{1}), ({4},{2,3},{1,3}), ({3,4},{2},{1}), ({3,4},{2},{1,4}), ({3,4},{2,4},{2,3}), ({3,4},{2,4},{1}), ({3,4},{2,4},{1,4}), ({3,4},{2,4},{1,3}), ({3,4},{2,4},{1,2}), ({3,4},{2,4},{1,2,3}), ({3,4},{1,2},{2,4}), ({3,4},{1,2,4},{2,3}), ({3,4},{1,2,4},{1,2,3}), ({2,3,4},{1,4},{1,3}), ({2,3,4},{1,4},{1,2,3}), ({2,3,4},{1,3,4},{1,2}), ({2,3,4},{1,3,4},{1,2,4}).
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- K. S. Brown, Dedekind's problem
- V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (in Russian), Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
- V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, 9, (1999), no. 6.
- G. Kilibarda and V. Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.
- Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
Crossrefs
Programs
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Magma
[n*(n-2)*(n-1)*(n+1)*(n^3 + 30*n^2 + 131*n - 270)/5040: n in [0..30]]; // G. C. Greubel, Nov 22 2017
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Mathematica
Table[Binomial[n+7,7]-6Binomial[n+5,5]+6Binomial[n+4,4]+3Binomial[n+3,3]- 6Binomial[n+2,2]+ 2Binomial[n+1,1],{n,0,40}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{0,0,0,2,19,90,302,820},40] (* Harvey P. Dale, Jul 27 2011 *) -
PARI
x='x+O('x^50); concat([0,0,0], Vec(x^3*(2+3*x-6*x^2+2*x^3)/(1-x)^8)) \\ G. C. Greubel, Oct 06 2017
Formula
a(n) = C(n+7, 7) - 6*C(n+5, 5) + 6*C(n+4, 4) + 3*C(n+3, 3) - 6*C(n+2, 2) + 2*C(n+1, 1).
a(n) = n*(n-2)*(n-1)*(n+1)*(n^3 + 30*n^2 + 131*n - 270)/5040.
G.f.: 1/(1-x)^8 - 6/(1-x)^6 + 6/(1-x)^5 + 3/(1-x)^4 - 6/(1-x)^3 + 2/(1-x)^2.
G.f.: x^3*(2 + 3*x - 6*x^2 + 2*x^3)/(1-x)^8.
Recurrence: a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8).
Generally, recurrence for the number of m-element ordered antichains on an unlabeled n-element set is a(m, n) = C(2^m, 1)*a(m, n - 1) - C(2^m, 2)*a(m, n - 2) + C(2^m, 3)*a(m, n - 3) + ... + ( - 1)^(k - 1)*C(2^m, k)*a(m, n - k) + ... - a(m, n - 2^m).
Extensions
More terms from Harvey P. Dale, Jul 27 2011
Comments