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%I #54 Jan 29 2023 17:51:09
%S 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,
%T 27,19,6,4,1,1,1,1,5,10,47,131,472,1326,3779,9013,19963,38073,65664,
%U 98804,133576,158658,169112,158658,133576,98804,65664,38073,19963,9013,3779,1326,472,131,47,10,5,1,1
%N 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.
%C 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]
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 112.
%D M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 150.
%H Jan Brandts, A. Cihangir, <a href="http://arxiv.org/abs/1512.03044">Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group</a>, arXiv preprint arXiv:1512.03044 [math.CO], 2015. See Fig. 13.
%H D. Condon, S. Coskey, L. Serafin, and C. Stockdale, <a href="http://arxiv.org/abs/1412.4683">On generalizations of separating and splitting families</a>, arXiv preprint arXiv:1412.4683 [math.CO], 2014-2015.
%H Jacob Feldman, <a href="http://ruccs.rutgers.edu/~jacob/Papers/feldman_catalog.pdf">A catalog of Boolean concepts</a>, Journal of Mathematical Psychology, Volume 47, Issue 1, 2003, 75-89.
%H Harald Fripertinger, <a href="http://dx.doi.org/10.1023/A:1008248618779">Enumeration, construction and random generation of block codes</a>, Designs, Codes, Crypt., 14 (1998), 213-219.
%H Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>
%H Harald Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/html2/construction/blockcodes_2.html">Enumeration of block codes</a>
%H Tilman Piesk, <a href="https://commons.wikimedia.org/wiki/File:Venn_and_Euler_diagram_of_3-ary_Boolean_relations.svg">Illustration of row 3</a>
%H <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>
%F Reference gives g.f.
%F Fripertinger gives g.f. for the number of classes of (n, m) nonlinear codes over an alphabet of size A.
%e Triangle begins:
%e k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 sums
%e n
%e 0 1 1 2
%e 1 1 1 1 3
%e 2 1 1 2 1 1 6
%e 3 1 1 3 3 6 3 3 1 1 22
%e 4 1 1 4 6 19 27 50 56 74 56 50 27 19 6 4 1 1 402
%t P = IntegerPartitions;
%t 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};
%t AC /@ Range[0, 5] // Flatten (* _Jean-François Alcover_, Dec 15 2019, after _Robert A. Russell_ in A034189 *)
%t 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 *)
%Y Row sums give A000616. Cf. A052265.
%Y Rows give A034188, A034189, A034190, etc.
%Y For other versions of this triangle see A171876, A039754, A276777.
%Y Cf. A171871. [_Robert Munafo_, Jan 25 2010]
%K nonn,tabf,nice
%O 0,8
%A _N. J. A. Sloane_
%E Corrected and extended by _Vladeta Jovovic_, Apr 20 2000
%E Entry revised by _N. J. A. Sloane_, Sep 19 2016
%E T(0, 1) = 1 inserted. (There are two 0-ary functions.) - _Tilman Piesk_, Jan 10 2023
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