A076836 -id:A076836 - cp's OEIS frontend

A034254 Triangle read by rows giving T(n,k) = number of inequivalent indecomposable linear [ n,k ] binary codes without 0 columns (n >= 2, 1 <= k <= n).

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 1, 1, 4, 10, 10, 4, 1, 1, 5, 18, 28, 18, 5, 1, 1, 7, 31, 71, 71, 31, 7, 1, 1, 8, 51, 165, 250, 165, 51, 8, 1, 1, 10, 79, 361, 809, 809, 361, 79, 10, 1, 1, 12, 121, 754, 2484, 3759, 2484, 754, 121, 12, 1, 1, 14, 177, 1503, 7240, 16749, 16749, 7240, 1503, 177, 14, 1

Offset: 1

N. J. A. Sloane

Fripertinger and Kerber (1995) mention that Slepian (1960) gave a generating function scheme for computing R_{n,k,2} = T(n,k), but it is not always correct. In Theorem 3.1, they give a corrected formula, but it seems too difficult to implement it in Sage. They do provide, however, a SYMMETRICA program for its computation (see the links). - Petros Hadjicostas, Oct 07 2019

			Triangle T(n,k) (with rows n >= 2 and columns k >= 1) begins as follows:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,   1;
  1, 3,  5,   3,   1;
  1, 4, 10,  10,   4,   1;
  1, 5, 18,  28,  18,   5,  1;
  1, 7, 31,  71,  71,  31,  7, 1;
  1, 8, 51, 165, 250, 165, 51, 8, 1;
  ...

Discrete algorithms at the University of Bayreuth, Symmetrica.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Rnk2: Number of the isometry classes of all binary indecomposable (n,k)-codes without zero columns. [This is a rectangular array, denoted by R_{nk2}, whose lower triangle (starting at n = 2) contains the current array T(n,k). The element R_{n=1,k=1,2} = 1 does not appear in the current array T(n,k).]
Harald Fripertinger, Enumeration of isometry-classes of linear (n,k)-codes over GF(q) in SYMMETRICA, Bayreuther Mathematische Schriften 49 (1999), 215-223. [For a SYMMETRICA program for the calculation of R_{nk2} = T(n,k), see pp. 219-220.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes, preprint, 1995. [We have T(n,k) = R_{nk2}; see p. 4 of the preprint.]
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. [We have T(n,k) = R_{nk2}; see p. 197.]
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.
Index entries for sequences related to binary linear codes

Cf. A076836 (row sums), A034253.

Columns include A000012 (k=1), A069905 (k=2), A034350 (k=3), A034351 (k=4), A034352 (k=5), A034353 (k=6), A034354 (k=7), A034355 (k=8).

More terms from Petros Hadjicostas, Oct 07 2019

A156802 Number of equivalence classes of connected bipartite graphs on n nodes up to sequences of edge local complementation and isomorphism.

Original entry on oeis.org

1, 1, 1, 2, 3, 8, 15, 43, 110, 370, 1260, 5366, 25684, 154104, 1156716

Offset: 1

Lars Eirik Danielsen (larsed(AT)ii.uib.no), Feb 16 2009

Also equal to the number of inequivalent indecomposable binary linear codes of length n plus the number of inequivalent indecomposable isodual binary linear codes of length n, divided by two.

L. E. Danielsen and M. G. Parker, Edge local complementation and equivalence of binary linear codes, Des. Codes Cryptogr., 49 (2008), 161-170.
L. E. Danielsen, Database of Pivot Orbits.

A076836

cp's OEIS Frontend

A034254 Triangle read by rows giving T(n,k) = number of inequivalent indecomposable linear [ n,k ] binary codes without 0 columns (n >= 2, 1 <= k <= n).

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