A034328 -id:A034328 - cp's OEIS frontend

1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 6, 16, 22, 16, 6, 1, 7, 23, 43, 43, 23, 7, 1, 8, 32, 77, 106, 77, 32, 8, 1, 9, 43, 131, 240, 240, 131, 43, 9, 1, 10, 56, 213, 516, 705, 516, 213, 56, 10, 1, 11, 71, 333, 1060, 1988, 1988, 1060, 333, 71, 11, 1, 12, 89

Offset: 1

N. J. A. Sloane

			Table T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  2,  1;
  3,  3,  1;
  4,  6,  4,   1;
  5, 10, 10,   5,  1;
  6, 16, 22,  16,  6,  1;
  7, 23, 43,  43, 23,  7, 1;
  8, 32, 77, 106, 77, 32, 8, 1;
  ...

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, preprint, 1995. [We have T(n,k) = W_{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) = 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(3) (2005), 691-699. [This paper apparently corrects errors in previous papers.]
Index entries for sequences related to binary linear codes

This is A076831 with the k=0 column omitted.
Columns include A000027 (k=1), A034198 (k=2), A034357 (k=3), A034358 (k=4), A034359 (k=5), A034360 (k=6), A034361 (k=7), A034362 (k=8).
Cf. A034253, A034254, A034328.

# Fripertinger's method to find the g.f. of column k >= 2 (for small k):
def A034356col(k, length):
    R = PowerSeriesRing(ZZ, 'x', default_prec=length)
    x = R.gen().O(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.list()
# For instance the Taylor expansion for column k = 4 gives
print(A034356col(4, 30)) # Petros Hadjicostas, Oct 07 2019

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=1: x/(1-x)^2.
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. For some cases, see also the links above.
(End)

A034327 Triangle of numbers of connected regular (k X n)-matrix matroids of dimension k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 3, 1, 1, 4, 9, 9, 4, 1, 1, 5, 17, 24, 17, 5, 1, 1, 7, 29, 60, 60, 29, 7, 1, 1, 8, 47, 133, 192, 133, 47, 8, 1, 1, 10, 72, 279, 556, 556, 279, 72, 10, 1, 1, 12, 110, 556, 1514, 2108, 1514, 556, 110, 12, 1

Offset: 0

N. J. A. Sloane

			The triangle begins:
  1;
  1;
  1, 1;
  1, 1, 1;
  1, 2, 2, 1;
  ...

Computed by Harald Fripertinger (fripert(AT)kfunigraz.ac.at).

H. Fripertinger, Isometry Classes of Codes
H. Fripertinger, Regular matroids
Index entries for sequences related to binary linear codes
Index entries for sequences related to matroids

Cf. A034328.

Columns: A034329, A034330, A034331, A034332.

A076766 Number of inequivalent binary linear codes of length n. Also the number of nonisomorphic binary matroids on an n-set.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 68, 148, 342, 848, 2297, 6928, 24034, 98854, 503137, 3318732, 29708814, 374039266, 6739630253, 173801649708, 6356255181216, 326203517516704, 23294352980140884, 2301176047764925736, 313285408199180770635, 58638266023262502962716

Offset: 0

Marcel Wild (mwild(AT)sun.ac.za), Nov 14 2002

			a(2)=4 because there are four inequivalent linear binary 2-codes: {(0,0)}, {(0,0),(1,0)}, {(0,0),(1,1)}, {(0,0),(1,0),(0,1),(1,1)}. Observe that the codes {(0,0),(1,0)} and {(0,0),(0,1)} are equivalent because one arises from the other by a permutation of coordinates.

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

Jayant Apte and J. M. Walsh, Constrained Linear Representability of Polymatroids and Algorithms for Computing Achievability Proofs in Network Coding, arXiv preprint arXiv:1605.04598 [cs.IT], 2016-2017.
Suyoung Choi and Hanchul Park, Multiplication structure of the cohomology ring of real toric spaces, arXiv:1711.04983 [math.AT], 2017.
H. Fripertinger, Isometry Classes of Codes.
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007.
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
James Oxley, What is a Matroid?.
Gordon Royle and Dillon Mayhew, 9-element matroids.
D. Slepian, On the number of symmetry types of Boolean functions of n variables, Canadian J. Math. 5, (1953), 185-193.
D. Slepian, A class of binary signaling alphabets, Bell System Tech. J. 35 (1956), 203-234.
D. Slepian, Some further theory of group codes, Bell System Tech. J. 39 1960 1219-1252. (Row sums of Table II.)
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.]
Index entries for sequences related to binary linear codes

Row sums of triangle A076831. Cf. A034328, A055545.

Edited by N. J. A. Sloane, Nov 01 2007, at the suggestion of Gordon Royle.

A174743 Partial sums of A076766.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 131, 279, 621, 1469, 3766, 10694, 34728, 133582, 636719, 3955451, 33664265, 407703531, 7147333784, 180948983492, 6537204164708, 332740721681412, 23627093701822296, 2324803141466748032, 315610211340647518667, 58953876234603150481383

Offset: 0

Jonathan Vos Post, Nov 30 2010

nonn

Number of inequivalent binary linear codes of length <= n. Also the total number of nonisomorphic binary matroids on an k-set for all k <= n. The subsequence of primes is: 3, 7, 31, 131.

			a(14) = 1 + 2 + 4 + 8 + 16 + 32 + 68 + 148 + 342 + 848 + 2297 + 6928 + 24034 + 98854 + 503137 = 636719 is prime.

Cf. A076766, A076831, A034328, A055545.

cp's OEIS Frontend

A034333 Number of matroids: column 3 of A034328.

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A034334 Number of matroids: column 4 of A034328.

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A034335 Number of matroids: column 5 of A034328.

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A034336 Number of matroids: column 6 of A034328.

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A034356 Triangle read by rows giving T(n,k) = number of inequivalent linear [n,k] binary codes (n >= 1, 1 <= k <= n).

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A034327 Triangle of numbers of connected regular (k X n)-matrix matroids of dimension k.

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A076766 Number of inequivalent binary linear codes of length n. Also the number of nonisomorphic binary matroids on an n-set.

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A174743 Partial sums of A076766.

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