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 A076831 Triangle T(n,k) read by rows giving number of inequivalent binary linear [n,k] codes (n >= 0, 0 <= k <= n). 14
 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 16, 22, 16, 6, 1, 1, 7, 23, 43, 43, 23, 7, 1, 1, 8, 32, 77, 106, 77, 32, 8, 1, 1, 9, 43, 131, 240, 240, 131, 43, 9, 1, 1, 10, 56, 213, 516, 705, 516, 213, 56, 10, 1, 1, 11, 71, 333, 1060, 1988, 1988 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS "The familiar appearance of the first few rows [...] provides a good example of the perils of too hasty extrapolation in mathematics." - Slepian. The difference between this triangle and the one for which it can be so easily mistaken is A250002. - Tilman Piesk, Nov 10 2014. REFERENCES M. Wild, Enumeration of binary and ternary matroids and other applications of the Brylawski-Lucas Theorem, Preprint No. 1693, Tech. Hochschule Darmstadt, 1994 LINKS Harald Fripertinger, Isometry Classes of Codes. Harald Fripertinger, Wnk2: Number of the isometry classes of all binary (n,k)-codes. [This is a rectangular array whose lower triangle contains T(n,k).] H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Apparently, the notation for T(n,k) is W_{nk2}; see p. 197.] Petros Hadjicostas, Generating function for column k=4. Petros Hadjicostas, Generating function for column k=5. Petros Hadjicostas, Generating function for column k=6. Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1. David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252. David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252. Marcel Wild, Consequences of the Brylawski-Lucas Theorem for binary matroids, European Journal of Combinatorics 17 (1996), 309-316. Marcel Wild, The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids, Finite Fields and their Applications 6 (2000), 192-202. Marcel Wild, The asymptotic number of binary codes and binary matroids, SIAM  J. Discrete Math. 19 (2005), 691-699. [This paper apparently corrects some errors in previous papers.] FORMULA From Petros Hadjicostas, Sep 30 2019: (Start) T(n,k) = Sum_{i = k..n} A034253(i,k) for 1 <= k <= n. G.f. for column k=2: -(x^3 - x - 1)*x^2/((x^2 + x + 1)*(x + 1)*(x - 1)^4). G.f. for column k=3: -(x^12 - 2*x^11 + x^10 - x^9 - x^6 + x^4 - x - 1)*x^3/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x^2 + x + 1)^2*(x^2 + 1)*(x + 1)^2*(x - 1)^8). G.f. for column k >= 4: modify the Sage program below (cf. function f). It is too complicated to write it here. (See also some of the links above.) (End) EXAMPLE k    0   1   2   3    4    5    6    7    8   9  10  11        sum    n    0      1                                                           1    1      1   1                                                       2    2      1   2   1                                                   4    3      1   3   3   1                                               8    4      1   4   6   4    1                                         16    5      1   5  10  10    5    1                                    32    6      1   6  16  22   16    6    1                               68    7      1   7  23  43   43   23    7    1                         148    8      1   8  32  77  106   77   32    8    1                    342    9      1   9  43 131  240  240  131   43    9   1                848   10      1  10  56 213  516  705  516  213   56  10   1           2297   11      1  11  71 333 1060 1988 1988 1060  333  71  11   1       6928 PROG (Sage) # Fripertinger's method to find the g.f. of column k >= 2 (for small k): def A076831col(k, length):     G1 = PSL(k, GF(2))     G2 = PSL(k-1, GF(2))     D1 = G1.cycle_index()     D2 = G2.cycle_index()     f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)     f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)     f = (f1 - f2)/(1-x)     return f.taylor(x, 0, length).list() # For instance the Taylor expansion for column k = 4 gives print(A076831col(4, 30)) # Petros Hadjicostas, Sep 30 2019 CROSSREFS Cf. A006116, A022166, A076766 (row sums). A034356 gives same table but with the k=0 column omitted. Columns include A000012 (k=0), A000027 (k=1), A034198 (k=2), A034357 (k=3), A034358 (k=4), A034359 (k=5), A034360 (k=6), A034361 (k=7), A034362 (k=8). Sequence in context: A130595 A108363 A329052 * A197061 A230861 A119724 Adjacent sequences:  A076828 A076829 A076830 * A076832 A076833 A076834 KEYWORD nonn,tabl,nice AUTHOR N. J. A. Sloane, Nov 21 2002 STATUS approved

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Last modified December 6 04:14 EST 2019. Contains 329784 sequences. (Running on oeis4.)