A051112 Number of monotone Boolean functions of n variables with 4 mincuts. Also Sperner systems with 4 blocks.
0, 0, 0, 0, 25, 2020, 82115, 2401910, 58089465, 1245331920, 24625121455, 460316430970, 8266174350005, 144171200793620, 2461016066613195, 41343340015862430, 686274244801356145, 11289648429330100120
Offset: 0
References
- J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, #8, s(n,4).
- V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean function, Belgrade, 1999, in preparation.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 0..831 (next term has 1001 digits)
- K. S. Brown, Dedekind's Problem
- D. M. Cvetkovic, The number of antichains of finite power sets, Publ. Inst. Math., 13 (27), 1972, 5-9.
- Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
- Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
- Index entries for linear recurrences with constant coefficients, signature (82, -2970, 62700, -856713, 7947786, -51019100, 226259000, -678011136, 1304341632, -1445575680, 696729600).
- Index entries for sequences related to Boolean functions
Programs
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Mathematica
Table[(1/4!)*(16^n-12*12^n+24*10^n+4*9^n-18*8^n+6*7^n-36*6^n+36*5^n+11*4^n-22*3^n+6*2^n),{n,0,20}] (* or *) LinearRecurrence[{82,-2970,62700,-856713,7947786,-51019100,226259000,-678011136,1304341632,-1445575680,696729600},{0,0,0,0,25,2020,82115,2401910,58089465,1245331920,24625121455},20] (* Harvey P. Dale, Nov 26 2019 *) -
PARI
a(n)=(16^n-12*12^n+24*10^n+4*9^n-18*8^n+6*7^n-36*6^n+36*5^n+11*4^n -22*3^n+6*2^n)/24 \\ Charles R Greathouse IV, Mar 14 2012
Formula
a(n) = (1/4!)*(16^n - 12*12^n + 24*10^n + 4*9^n - 18*8^n + 6*7^n - 36*6^n + 36*5^n + 11*4^n - 22*3^n + 6*2^n).
From Michael Somos: (Start)
a(n) = 82*a(n - 1) - 2970*a(n - 2) + 62700*a(n - 3) - 856713*a(n - 4) + 7947786*a(n - 5) - 51019100*a(n - 6) + 226259000*a(n - 7) - 678011136*a(n - 8) + 1304341632*a(n - 9) - 1445575680*a(n - 10) + 696729600*a(n - 11).
G.f.: 5x^4(5-6x-1855x^2+20076x^3-44356x^4-215280x^5+759168x^6) / ((1-3x)(1-4x)(1-5x)(1-6x)(1-2x)(1-7x)(1-8x)(1-9x)(1-10x)(1-12x)(1-16x)). (End)