A034343 -id:A034343 - cp's OEIS frontend

A076832 Triangle T(n,k), read by rows, giving the total number of inequivalent binary linear [n,i] codes with no column of zeros, summed for i <= k (n >= 1, 1 <= k <= n).

1, 1, 2, 1, 3, 4, 1, 4, 7, 8, 1, 5, 11, 15, 16, 1, 7, 19, 30, 35, 36, 1, 8, 29, 56, 73, 79, 80, 1, 10, 44, 107, 161, 186, 193, 194, 1, 12, 66, 200, 363, 462, 497, 505, 506, 1, 14, 96, 372, 837, 1222, 1392, 1439, 1448, 1449, 1, 16, 136, 680, 1963, 3435, 4282

Offset: 1

N. J. A. Sloane, Nov 21 2002

From Petros Hadjicostas, Sep 30 2019: (Start)
It seems that Harald Fripertinger at his website defines T(n,k) = T(n,n) for k > n (and thus he gets an orthogonal array). It seems that T(n,n) = A034343(n).
It seems that T(n,k=2) = A001399(n) for n >= 2 (with A001399(n=1) = T(1,1)); T(n,k=3) = A034337(n) for n >= 3 (with A034337(n) = T(n,n) for 1 <= n <= 2); T(n,k=4) = A034338(n) for n >= 4 (with A034338(n) = T(n,n) for 1 <= n <= 3); and so on. See the Crossrefs below for more information.
To get the g.f. of column k (starting at n = 0 with T(n=0,k) := 1 rather than at n = k), modify the Sage program below (cf. function f).
(End)

			Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1,  2;
  1,  3,  4;
  1,  4,  7,   8;
  1,  5, 11,  15,  16;
  1,  7, 19,  30,  35,  36;
  1,  8, 29,  56,  73,  79,  80;
  1, 10, 44, 107, 161, 186, 193, 194; ...

Discrete algorithms at the University of Bayreuth, Symmetrica. [This package was used to compute T_{nk2} using the cycle index of PGL_k(2).]
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Tnk2: Number of the isometry classes of all binary (n,r)-codes for 1 <= r <= k without zero-columns. [This is a rectangular array whose lower triangle contains T(n,k).]
Harald Fripertinger, Enumeration of isometry classes of linear (n,k)-codes over GF(q) in SYMMETRICA, Bayreuther Mathematische Schriften 49 (1995), 215-223. [See pp. 216-218. A C-program is given for calculating T_{nk2} in Symmetrica.]
Harald Fripertinger, Cycle of indices of linear, affine, and projective groups, Linear Algebra and its Applications 263 (1997), 133-156. [See p. 152 for the computation of T_{nk2}.]
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 T_{nk2}.]
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.
Wikipedia, Cycle index.
Wikipedia, Projective linear group.
Index entries for sequences related to binary linear codes

Columns give truncated versions of A001399 (k = 2), A034337 (k = 3), A034338 (k = 4), A034339 (k = 5), A034340 (k = 6), A034341 (k = 7), A034342 (k = 8), and A034343 (? main diagonal).

Cf. A034253, A076831.

# We illustrate how to get a g.f. for column k in Maple when k is small.
with(GroupTheory);
G := ProjectiveGeneralLinearGroup(4, 2);
GroupOrder(G);
# We get that the order is 20160.
f:=CycleIndexPolynomial(G, [x||(1..20160)]);
# We get
# 1/20160*x1^15 + 1/192*x1^7*x2^4 + 1/96*x1^3*x2^6 + 1/16*x1^3*x2^2*x4^2 +
# 1/18*x1^3*x3^4 + 1/6*x1*x2*x3^2*x6 + 1/8*x1*x2*x4^3 + 1/180*x3^5 + 2/7*x1*x7^2 +
# 1/12*x3*x6^2 + 1/15*x5^3 + 2/15*x15
# The only dummy variables that appear are x1, x2, x3, x4, x5, x6, x7, and x15.
g:=subs(x1 = 1/(1 - y), subs(x2 = 1/(-y^2 + 1), subs(x3 = 1/(-y^3 + 1), subs(x4 = 1/(-y^4 + 1), subs(x5 = 1/(-y^5 + 1), subs(x6 = 1/(-y^6 + 1), subs(x7 = 1/(-y^7 + 1), subs(x15 = 1/(-y^15 + 1), f))))))));
# Then we take a Taylor expansion of the above g.f.
taylor(g,y=0,50);
# We get a Taylor expansion for column k = 4 (i.e., A034338).
# Petros Hadjicostas, Sep 30 2019

# Fripertinger's method to find the g.f. of column k for small k:
def A076832col(k, length):
    G = PSL(k, GF(2))
    D = G.cycle_index()
    f = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D)
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 4 gives A034338:
print(A076832col(4, 30)) # Petros Hadjicostas, Sep 30 2019

Revised by N. J. A. Sloane, Mar 01 2004

A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

Original entry on oeis.org

1, 3, 6, 15, 24, 60, 105, 255, 384, 960, 1632, 1680, 4080, 15555, 27030, 65535, 98304, 245760, 417792, 430080, 1044480, 1582080, 3947520, 3982080, 6908160, 6919680, 16776960, 106991625, 267448335, 1019462460, 1771476585, 4294967295

Offset: 0

Tilman Piesk, Aug 01 2013

A subsequence of A227723, showing all the big equivalence classes that contain Boolean functions related to subgroups of nimber addition (A190939).
Forms a triangle with row lengths A034343 = 1, 1, 2, 4, 8, 16, 36, 80...:
1,
3,
6, 15,
24, 60, 105, 255,
384, 960, 1632, 1680, 4080, 15555, 27030, 65535...
The left column a( 1,2,4,8,16,32,68,148... ) = a( A076766 ) = 3 ,6, 24, 384, 98304... is probably A001146 * 3/2, which is also A006017( A000079 ).
The first A076766(n) entries correspond to the first A006116(n) entries of A190939. (The first 148 here, for n = 7,  correspond to the first 29212 there.) The entries of A190939 can be generated from this sequence.
Among the first A076766(n) entries are A076831(n;0...n) with weight 2^0...2^n. (Among the first 148 are 1, 7, 23, 43, 43, 23, 7, 1 with weights 1, 2, 4, 8, 16, 32, 64, 128.)
a(n) appears to be divisible by 3 for n>0, and the odd part of a(n) is almost always squarefree. - Ralf Stephan, Aug 02 2013

Tilman Piesk, Table of n, a(n) for n = 0..147
Tilman Piesk, a(n) for n = 0..147 and corresponding entries of A190939 in reverse binary. Prime factors and numbers of prime factors (A001222).
Tilman Piesk, Subgroups of nimber addition (Wikiversity)

Subsequence of A227723 (all becs). All entries are also in A227963 (all sona-secs). Neither shares the property of divisibility by 3.
A190939, A034343, A076766, A076831.
The prime factors contain many prime factors of Fermat numbers (A023394).

a( A076766 - 1 ) = A001146 - 1 = A051179.

a( A076766 ) = A001146 * 3/2 (probably).

A076893 Number of inequivalent linear ternary codes of length n with no zero columns. Also the number of nonisomorphic loopless ternary matroids on an n-set.

Original entry on oeis.org

1, 2, 4, 9, 19, 49, 131, 424, 1652, 8719, 69825, 989873, 26581301, 1318262256, 114250250466, 16765968053831, 4100014001380089, 1664142340751871950, 1116818491052884840894, 1241182654624957299191014, 2281173954695869520832771726, 6954114729274902830973149757397, 35138195971007372542783032580880899

Offset: 1

Marcel Wild (mwild(AT)sun.ac.za), Nov 26 2002

M. Wild, Enumeration of binary and ternary matroids and other applications of the Brylawski-Lucas Theorem, Preprint 1693, Technische Hochschule Darmstadt, 1994.

A. Evnin, An elementary Introduction to Matroids, Mathematical Education 2 (33) (2005), 2-33.
Harald Fripertinger, T_nk3: Number of the isometry classes of all ternary (n,r)-codes for 1 <= r <= k without zero-columns, 2016.

Cf. A034343.

a(11)-a(23) from Fripertinger's table added by Andrei Zabolotskii, Aug 26 2025

cp's OEIS Frontend

A007669 Duplicate of A034343.

Views

Author

Keywords

A076832 Triangle T(n,k), read by rows, giving the total number of inequivalent binary linear [n,i] codes with no column of zeros, summed for i <= k (n >= 1, 1 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Sage

Extensions

A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A076893 Number of inequivalent linear ternary codes of length n with no zero columns. Also the number of nonisomorphic loopless ternary matroids on an n-set.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A153447 Duplicate of A076893.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A156803 Number of equivalence classes of bipartite graphs on n nodes up to sequences of edge local complementation and isomorphism.

Views

Author

Keywords

Comments

Links

Crossrefs