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0.2.1.6

On investigations regarding categories of clausesets. More...
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On investigations regarding categories of clausesets.
prl(L) := product_flcls(map(fcls2flcls, map(full_fcs,L)))$ prlfcs(L) := flcls2fcls(prl(L))$ testl(L) := current_satsolver(prlfcs(L))$ testl([0,0]); false; testl([0,1]); true; testl([1,1]); false; testl([1,2]); true; testl([1,3]); true; testl([1,4]); true; testl([1,5]); true; testl([1,6]); true; prl([1,2]); [ [{[1,2],[1,1],[1,1],[1,2],[1,2],[1,1],[1,1],[1,2]}, lambda([x],map('apply,[lambda([x],x),lambda([x],x)],map("[",x)))], {{[1,2],[1,1]},{[1,2],[1,1]},{[1,1],[1,2]},{[1,1],[1,2]}, {[1,2],[1,1]},{[1,2],[1,1]},{[1,1],[1,2]},{[1,1],[1,2]}} ] flcls2fcls(prl([1,2])); [{1,2,3,4},{{4,3},{4,2},{3,1},{2,1},{1,2},{1,3},{2,4},{3,4}}] all_sat_fcs(flcls2fcls(prl([1,2]))); {{4,1,2,3},{3,2,1,4}} % Solutions interpreted: [1,1],[1,1],[1,2],[1,2] 2,3,4,1 [1,1],[1,1],[1,2],[1,2] 2,3,4,1 length(all_sat_fcs(flcls2fcls(prl([1,2])))); 2 length(all_sat_fcs(flcls2fcls(prl([1,3])))); 18 length(all_sat_fcs(flcls2fcls(prl([1,4])))); 110 length(all_sat_fcs(flcls2fcls(prl([1,5])))); 570 length(all_sat_fcs(flcls2fcls(prl([1,6])))); 2702 length(all_sat_fcs(flcls2fcls(prl([1,7])))); 12138 map(factor,[2,18,110,570,2702]); [2, 2*3^2, 2*5*11, 2*3*5*19, 2*7*193] eis_search(2,18,110,570,2702); ["A038721"] eis_search_name(2,18,110,570,2702); A038721  k=2 column of A038719. eis_search(1,9,55,285,1351); ["A016269"] eis_search_name(1,9,55,285,1351); A016269  Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks. eis_details(A016269); A016269  Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks. UNSIGNED TERMS 1, 9, 55, 285, 1351, 6069, 26335, 111645, 465751, 1921029, 7859215, 31964205, 129442951,522538389, 2104469695, 8460859965, 33972448951, 136276954149, 546269553775, 2188563950925, 8764714059751 OFFSET 0, 2 FORMULA G.f.: 1/((12x)(13x)(14x)). a(n) = (2^n)*(2^n  1)/2  3^n + 2^n. a(n)=sum{0<=i,j,k,<=n, i+j+k=n, 2^i*3^j*4^k}.  Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007 a(n)=2^(n+1)*(1+2^(n+2))3^(n+2).  Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 25 2007 a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3).  Ross La Haye (rlahaye(AT)new.rr.com), Jan 10 2008 MAPLE PROGRAM with(combinat):a:=n>stirling2(n,4)stirling2(n1,4): seq(a(n), n=4..24);  Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 05 2007 CROSSREFERENCES Equals (1/2) A038721(n+1). First differences of A000453. Partial sums of A027650. Pairwise sums of A099110. Odd part of A019333. Cf. A000392, A032263. Adjacent sequences: A016266 A016267 A016268 this_sequence A016270 A016271 A016272 Sequence in context: A058852 A068970 A141530 this_sequence A005770 A030053 A072844 REFERENCES L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 292, #8, s(n,2). HYPERLINKS K. S. Brown, Dedekind's problem [http://www.mathpages.com/home/kmath030.htm] Vladeta Jovovic, Illustration for A016269, A047707, A051112A051118 [http://www.research.att.com/~njas/sequences/a047707.pdf] Index entries for sequences related to Boolean functions [http://www.research.att.com/~njas/sequences/Sindx_Bo.html#Boolean] Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets [http://www.cs.uwaterloo.ca/journals/JIS/index.html], J. Integer Seqs., Vol. 7, 2004. COMMENTS Half the number of 2 X (n+2) binary arrays with both a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.  Ron Hardin (rhh(AT)cadence.com), Mar 21 2002 As (0,0,1,9,55,...) this is the third binomial transform of cosh(x)1. It is the binomial transform of A000392, when this has two leading zeros. Its e.g.f. is then exp(3x)cosh(x)exp(3x) and a(n)=(4^n2*3^n+2^n)/2.  Paul Barry (pbarry(AT)wit.ie), May 13 2003 Let P(A) be the power set of an nelement set A. Then a(n2) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x.  Ross La Haye (rlahaye(AT)new.rr.com), Jan 10 2008 testl(2,2); false; testl(2,3); false; testl(3,3); false; testl([2,2,2]); false; testl([2,2,3]); false; testl([2,2,2,2]);
map(bipartite_fcs_p,create_list(prlfcs([1,n]),n,1,5)); [true,true,true,true,true] map(lambda([FF], is(comp_fcs(FF) = FF)), create_list(prlfcs([1,n]),n,1,5)); [true,true,true,true,true]
is_isomorphic_btr_fcs(tcol2sat_hg2fcs(fcs2hg(full_fcs(1))), prlfcs([1,1])); true is_isomorphic_btr_fcs(tcol2sat_hg2fcs(fcs2hg(full_fcs(2))), prlfcs([1,2])); true is_isomorphic_btr_fcs(tcol2sat_hg2fcs(fcs2hg(full_fcs(3))), prlfcs([1,3])); true is_isomorphic_btr_fcs(tcol2sat_hg2fcs(fcs2hg(full_fcs(4))), prlfcs([1,4])); true is_isomorphic_btr_fcs(tcol2sat_hg2fcs(fcs2hg(full_fcs(5))), prlfcs([1,5])); true
alldisjnepairs(n) := second(kneser_g_hg(ses2hg(disjoin({},powerset(setn(n)))))); an_v1(n) := sum_l(map( lambda([P], 2^(nlength(first(P))length(second(P)))), listify(alldisjnepairs(n)))); create_list(an_v1(n),n,1,6); [0,1,9,55,285,1351]
all_Sp_2bl(n) := subset(powerset(powerset(setn(n)), 2), lambda([S], is(length(S)=2) and not elementp({},S) and is_antichain(S)))$ an_v2(n) := length(all_Sp_2bl(n))$ create_list(an_v2(n),n,1,6); [0,1,9,55,285,1351]
an_v3(n) := binomial(2^n,1) + binomial(2^n,2)  3^n; create_list(an_v3(n),n,1,10); [0,1,9,55,285,1351,6069,26335,111645,465751]
Definition in file general.hpp.