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A039754 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. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
For N=1 through N=5, the first 2^(N-1) terms of row N are also found in triangle A171871, which is related to A005646. This was shown for all N by Andrew Weimholt, Dec 30 2009. [Robert Munafo, Jan 25 2010]
REFERENCES
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.
LINKS
Jan Brandts, A. Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015. See Fig. 13.
D. Condon, S. Coskey, L. Serafin, and C. Stockdale, On generalizations of separating and splitting families, arXiv preprint arXiv:1412.4683 [math.CO], 2014-2015.
Jacob Feldman, A catalog of Boolean concepts, Journal of Mathematical Psychology, Volume 47, Issue 1, 2003, 75-89.
Harald Fripertinger, Enumeration, construction and random generation of block codes, Designs, Codes, Crypt., 14 (1998), 213-219.
Harald Fripertinger, Isometry Classes of Codes
Harald Fripertinger, Enumeration of block codes
Tilman Piesk, Illustration of row 3
FORMULA
Reference gives g.f.
Fripertinger gives g.f. for the number of classes of (n, m) nonlinear codes over an alphabet of size A.
EXAMPLE
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
MATHEMATICA
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};
AC /@ Range[0, 5] // Flatten (* Jean-François Alcover, Dec 15 2019, after Robert A. Russell in A034189 *)
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 *)
CROSSREFS
Row sums give A000616. Cf. A052265.
Rows give A034188, A034189, A034190, etc.
For other versions of this triangle see A171876, A039754, A276777.
Cf. A171871. [Robert Munafo, Jan 25 2010]
Sequence in context: A129179 A120621 A201080 * A213919 A362043 A337220
KEYWORD
nonn,tabf,nice
AUTHOR
EXTENSIONS
Corrected and extended by Vladeta Jovovic, Apr 20 2000
Entry revised by N. J. A. Sloane, Sep 19 2016
T(0, 1) = 1 inserted. (There are two 0-ary functions.) - Tilman Piesk, Jan 10 2023
STATUS
approved

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Last modified June 13 10:05 EDT 2024. Contains 373383 sequences. (Running on oeis4.)