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A132581
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Number of antichains in the first n elements of the infinite Boolean lattice.
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4
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1, 2, 3, 5, 6, 11, 14, 19, 20, 39, 53, 78, 84, 134, 148, 167, 168, 335, 483, 765, 849, 1466, 1681, 1988, 2008, 3700, 4414, 5489, 5573, 7265, 7413, 7580, 7581, 15161, 22574, 37252, 42825, 77388, 92864, 116454, 118462, 227503, 286776, 382574, 392247, 555662, 574114, 595481, 595649, 1176304, 1563955
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OFFSET
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0,2
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COMMENTS
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Every nonnegative integer represents a finite set of nonnegative integers and conversely, by the mapping that takes n into the exponents of 2 in its binary representation. Thus 0 represents the empty set and 9 represents {0,3}, etc.
The sequence [more precisely a(n+1) - Ed.] counts the antichains in the partial ordering {0,1,...,n}, which is really the family of sets {emptyset,{0},{1},{0,1},{2},...}.
The sequence of differences, A132582, enumerates the antichains in the infinite Boolean lattice {0,1,2,...} whose largest element is n [for A132582(n) = a(n+1) - a(n)]. For example, the five such antichains when n = 6 are {6}, {1,6}, {3,6}, {5,6}, {3,5,6}.
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.1.4.
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LINKS
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FORMULA
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a(2^n) = a(2^n-2^m) + a(2^n-2^(n-m)) for n >= m >= 0.
a(2^n+2) = a(2^n) + a(2^n-1) + a(2^n-2) = 3*a(2^n-2) + a(2^(n-1)+1) for n >= 1.
a(2^n+1) = 2*a(2^n-2) + a(2^(n-1)+1) = 2*a(2^n-1) + 1 for n >= 1.
a(2^n-1) = a(2^n-2) + a(2^(n-1)-1) for n >= 1.
a(2^n-2) = a(2^n-3) + a(2^(n-1)-2) for n >= 2.
(End)
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EXAMPLE
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For n = 0, we consider the first zero elements of the lattice, i.e., no element at all. Thus a(0) = 1 counts the only antichain with elements in the empty set, i.e., no elements, which is the empty antichain {}.
For n = 1, we consider the first element of the lattice, represented by the first nonnegative integer, 0: This element is the empty set. Hence a(1) = 2 counts the antichains with elements in { {} }, the singleton containing the empty set. These two antichains are again the empty antichain, and the singleton antichain containing just the empty set, { {} }.
For n = 2, we consider the first two elements of the lattice, represented by the integers 0 and 1, which are {} and {0}. So a(2) = 3 counts the antichains with elements in { {}, {0} }: These three antichains are the empty antichain {} and the two singleton antichains { {} } and { {0} }.
For n = 3, we consider the first three elements of the lattice, represented by 0, 1 and 2; these are the sets {}, {0}, {1}. Hence a(3) counts the 5 antichains with elements in { {}, {0}, {1} }: the empty antichain {}, and the three singleton antichains { {} }, { {0} }, { {1} }, and the antichain { {0}, {1} }.
For n = 4, the first 4 elements of the lattice, represented by 0, 1, 2 and 3, form the power set P({0, 1}) = { {}, {0}, {1}, {0, 1} }. Hence a(4) counts all 6 antichains of subsets of {0, 1}, which is equivalent to considering the antichains of subsets of {1, 2} as counted by A000372(2), see example there.
(End)
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MAPLE
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N:= 63:
Q:= [seq(convert(n+64, base, 2), n=0..N)]:
Incomp:= Array(0..N, 0..N, proc(i, j) local d; d:= Q[i+1]-Q[j+1]; has(d, 1) and has(d, -1) end proc):
AntichainCount:= proc(S) option cache; local t, r;
1 + add(procname(select(s -> Incomp[s, S[t]], S[1..t-1])) , t = 1..nops(S));
end proc:
seq(AntichainCount([$0..n]), n=-1..N);
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MATHEMATICA
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M = 63;
Q = Table[IntegerDigits[n + 64, 2], {n, 0, M}];
Incomp[i_, j_] := Module[{d}, d = Q[[i + 1]] - Q[[j + 1]]; MemberQ[d, 1] && MemberQ[d, -1]];
AntichainCount[S_] := AntichainCount[S] = Module[{t, r}, 1 + Sum[ AntichainCount[Select[S[[1 ;; t - 1]], Incomp[#, S[[t]]]&]], {t, 1, Length[S]}]];
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PROG
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(PARI) M132581 = Map(); A132581(n) = { if( mapisdefined(M132581, n, &n), n, n<3, n+1, my(e=exponent(n)); mapput(M132581, n, e = if( n==2^e, for( b=e\/2, e, iferr( e=A132581(n-2^b)+ A132581(n-n>>b); break, e, )); type(e)=="t_INT"|| error(e); e, n==2<<e - 1, A132581(2^e-1)+A132581(n-1), n==2<<e - 2, A132581(2^e-2) + A132581(n-1), n==2^e + 1, iferr( A132581(n\/2) + 2*A132581(n-3), e, 2*A132581(n-2) + 1), n==2^e + 2, iferr( A132581(n\2) + 3*A132581(n-4), e, A132581(n-4) + A132581(n-3) + A132581(n-2)), mapisdefined(M132581, [-n], &e), mapdelete(M132581, [-n]); mapdelete(M132581, [n]); error(e" without success."), mapisdefined(M132581, [n]), mapput(M132581, [-n], Str("Trying to use A132582("n")")); A132581(n+1)-A132582(n)-mapdelete(M132581, [-n]), mapput(M132581, [n], 0); A132581(n-1)+A132582(n-1)+mapdelete(M132581, [n]))); e)} \\ So far, not all values can be computed. - M. F. Hasler, Jun 04 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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