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A039754
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Irregular triangle read by rows: T(n,k) = number of binary codes of length n with k words (n >= 0, 0 <= k <= 2^n); also number of 0/1-polytopes with vertices from the unit n-cube; also number of inequivalent Boolean functions of n variables with exactly k nonzero values under action of Jevons group.
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15
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 6, 3, 3, 1, 1, 1, 1, 4, 6, 19, 27, 50, 56, 74, 56, 50, 27, 19, 6, 4, 1, 1, 1, 1, 5, 10, 47, 131, 472, 1326, 3779, 9013, 19963, 38073, 65664, 98804, 133576, 158658, 169112, 158658, 133576, 98804, 65664, 38073, 19963, 9013, 3779, 1326, 472, 131, 47, 10, 5, 1, 1
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OFFSET
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0,8
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COMMENTS
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REFERENCES
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F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 112.
M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 150.
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LINKS
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FORMULA
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Reference gives g.f.
Fripertinger gives g.f. for the number of classes of (n, m) nonlinear codes over an alphabet of size A.
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EXAMPLE
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Triangle begins:
k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 sums
n
0 1 1 2
1 1 1 1 3
2 1 1 2 1 1 6
3 1 1 3 3 6 3 3 1 1 22
4 1 1 4 6 19 27 50 56 74 56 50 27 19 6 4 1 1 402
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MATHEMATICA
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P = IntegerPartitions;
AC[d_Integer] := Module[{C, M, p}, (* from W. Y. C. Chen algorithm *) M[p_List] := Plus @@ p!/(Times @@ p * Times @@ (Length /@ Split[p]!)); C[p_List, q_List] := Module[{r, m, k, x}, r = If[0 == Length[q], 1, 2*2^IntegerExponent[LCM @@ q, 2]]; m = LCM @@ Join[p/GCD[r, p], q/GCD[r, q]]; CoefficientList[Expand[Product[(1 + x^(k *r))^((Plus @@ Map[MoebiusMu[k/#]*2^Plus @@ GCD[#*r, Join[p, q]]&, Divisors[k]])/(k*r)), {k, 1, m}]], x]]; Sum[Binomial[d, p]*Plus @@ Plus @@ Outer[M[#1] M[#2] C[#1, #2]*2^(d - Length[#1] - Length[#2]) &, P[p], P[d - p], 1], {p, 0, d}]/(d! 2^d)]; AC[0] = {1, 1};
Table[ CoefficientList[ CycleIndexPolynomial[ GraphData[ {"Hypercube", n}, "AutomorphismGroup"], Array[Subscript[x, ##] &, 2^n]] /. Table[ Subscript[x, i] -> 1 + x^i, {i, 1, 2^n}], x], {n, 1, 8}] // Grid (* Geoffrey Critzer, Jan 10 2020 *)
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CROSSREFS
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KEYWORD
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nonn,tabf,nice
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AUTHOR
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EXTENSIONS
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T(0, 1) = 1 inserted. (There are two 0-ary functions.) - Tilman Piesk, Jan 10 2023
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STATUS
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approved
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