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.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabf
• nice
• changed

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]

Examples

			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
		

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
• Index entries for sequences related to Boolean functions

Crossrefs

Row sums give A000616. Cf. A052265.

Rows give A034188, A034189, A034190, etc.

Columns are A034198, A034199, A034200, etc.

Diagonal is A276412.

For other versions of this triangle see A171876, A039754, A276777.

Cf. A171871.

Programs

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 *)

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.

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