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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

%I

%S 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,

%T 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,

%U 10,56,213,516,705,516,213,56,10,1,1,11,71,333,1060,1988,1988

%N Triangle T(n,k) read by rows giving number of inequivalent binary linear [n,k] codes (n >= 0, 0 <= k <= n).

%C "The familiar appearance of the first few rows [...] provides a good example of the perils of too hasty extrapolation in mathematics." - Slepian.

%C The difference between this triangle and the one for which it can be so easily mistaken is A250002. - _Tilman Piesk_, Nov 10 2014.

%D M. Wild, Enumeration of binary and ternary matroids and other applications of the Brylawski-Lucas Theorem, Preprint No. 1693, Tech. Hochschule Darmstadt, 1994

%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/codes/tables_6.html">Wnk2: Number of the isometry classes of all binary (n,k)-codes</a>. [This is a rectangular array whose lower triangle contains T(n,k).]

%H H. Fripertinger and A. Kerber, <a href="https://doi.org/10.1007/3-540-60114-7_15">Isometry classes of indecomposable linear codes</a>. 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.]

%H Petros Hadjicostas, <a href="/A076831/a076831.txt">Generating function for column k=4</a>.

%H Petros Hadjicostas, <a href="/A076831/a076831_1.txt">Generating function for column k=5</a>.

%H Petros Hadjicostas, <a href="/A076831/a076831_2.txt">Generating function for column k=6</a>.

%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

%H David Slepian, <a href="https://archive.org/details/bstj39-5-1219">Some further theory of group codes</a>, Bell System Tech. J. 39(5) (1960), 1219-1252.

%H David Slepian, <a href="https://doi.org/10.1002/j.1538-7305.1960.tb03958.x">Some further theory of group codes</a>, Bell System Tech. J. 39(5) (1960), 1219-1252.

%H Marcel Wild, <a href="http://dx.doi.org/10.1006/eujc.1996.0026">Consequences of the Brylawski-Lucas Theorem for binary matroids</a>, European Journal of Combinatorics 17 (1996), 309-316.

%H Marcel Wild, <a href="http://dx.doi.org/10.1006/ffta.1999.0273">The asymptotic number of inequivalent binary codes and nonisomorphic binary matroids</a>, Finite Fields and their Applications 6 (2000), 192-202.

%H Marcel Wild, <a href="https://doi.org/10.1137/S0895480104445538">The asymptotic number of binary codes and binary matroids</a>, SIAM J. Discrete Math. 19 (2005), 691-699. [This paper apparently corrects some errors in previous papers.]

%H <a href="/index/Coa#codes_binary_linear">Index entries for sequences related to binary linear codes</a>

%F From _Petros Hadjicostas_, Sep 30 2019: (Start)

%F T(n,k) = Sum_{i = k..n} A034253(i,k) for 1 <= k <= n.

%F G.f. for column k=2: -(x^3 - x - 1)*x^2/((x^2 + x + 1)*(x + 1)*(x - 1)^4).

%F 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).

%F 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.)

%F (End)

%e k 0 1 2 3 4 5 6 7 8 9 10 11 sum

%e n

%e 0 1 1

%e 1 1 1 2

%e 2 1 2 1 4

%e 3 1 3 3 1 8

%e 4 1 4 6 4 1 16

%e 5 1 5 10 10 5 1 32

%e 6 1 6 16 22 16 6 1 68

%e 7 1 7 23 43 43 23 7 1 148

%e 8 1 8 32 77 106 77 32 8 1 342

%e 9 1 9 43 131 240 240 131 43 9 1 848

%e 10 1 10 56 213 516 705 516 213 56 10 1 2297

%e 11 1 11 71 333 1060 1988 1988 1060 333 71 11 1 6928

%o (Sage) # Fripertinger's method to find the g.f. of column k >= 2 (for small k):

%o def A076831col(k, length):

%o G1 = PSL(k, GF(2))

%o G2 = PSL(k-1, GF(2))

%o D1 = G1.cycle_index()

%o D2 = G2.cycle_index()

%o f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)

%o f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)

%o f = (f1 - f2)/(1-x)

%o return f.taylor(x, 0, length).list()

%o # For instance the Taylor expansion for column k = 4 gives

%o print(A076831col(4, 30)) # _Petros Hadjicostas_, Sep 30 2019

%Y Cf. A006116, A022166, A076766 (row sums).

%Y A034356 gives same table but with the k=0 column omitted.

%Y 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).

%K nonn,tabl,nice

%O 0,5

%A _N. J. A. Sloane_, Nov 21 2002

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