A055545 -id:A055545 - cp's OEIS frontend

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

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.

A002773 Number of nonisomorphic simple matroids (or geometries) with n points.

Original entry on oeis.org

1, 1, 1, 2, 4, 9, 26, 101, 950, 376467

Offset: 0

N. J. A. Sloane

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 138.
Knuth, Donald E. "The asymptotic number of geometries." Journal of Combinatorial Theory, Series A 16.3 (1974): 398-400.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. Bansal, R. Pendavingh, and J. G. van der Pol, On the number of matroids, arXiv:1206.6270v1 [math.CO], 2012.
Nikhil Bansal, Rudi A. Pendavingh, and Jorn G. van der Pol, On the number of matroids, Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2013; full version in Combinatorica, 35:3 (2015), 253-277.
J. E. Blackburn, H. H. Crapo, and D. A. Higgs, A catalogue of combinatorial geometries, Math. Comp 27 (1973), 155-166.
Henry H. Crapo and Gian-Carlo Rota, On the foundations of combinatorial theory. II. Combinatorial geometries, Studies in Appl. Math. 49 (1970), 109-133.
Henry H. Crapo and Gian-Carlo Rota, On the foundations of combinatorial theory. II. Combinatorial geometries, Studies in Appl. Math. 49 (1970), 109-133. [Annotated scanned copy of pages 126 and 127 only]
W. M. B. Dukes, Tables of matroids.
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
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.
M. J. Piff, An upper bound for the number of matroids, J. Combinatorial Theory Ser. B, vol 14 (1973), pp. 241-245.
Gordon Royle and Dillon Mayhew, 9-element matroids.
N. J. A. Sloane, Initial terms (* denotes 5 points in general position in 4-space).
Eric Weisstein's World of Mathematics, Matroid.
Index entries for sequences related to matroids

Cf. A055545, A056642. Row sums of A058730.

Limit_{ n -> oo }  (log_2 log_2 a(n))/n = 1. [Knuth]
2^n/n^(3/2) << log a(n) << 2^n/n, proved by Knuth and Piff respectively. - Charles R Greathouse IV, Mar 20 2021
Bansal, Pendavingh, & van der Pol prove an upper bound almost matching the lower bound above: log a(n) <= 2*sqrt(2/Pi)*2^n/n^(3/2)*(1 + o(1)). - Charles R Greathouse IV, Mar 20 2021

a(9) from Gordon Royle, Dec 23 2006

A053534 Triangle T(n,k) giving number of pairwise non-isomorphic (i.e., unlabeled) matroids of rank k on n points (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 23, 38, 23, 6, 1, 1, 7, 37, 108, 108, 37, 7, 1, 1, 8, 58, 325, 940, 325, 58, 8, 1, 1, 9, 87, 1275, 190214, 190214, 1275, 87, 9, 1

Offset: 0

N. J. A. Sloane, Dec 30 2000

			The triangle, transposed, begins:
k...n=0...n=1...n=2...n=3...n=4...n=5...n=6...n=7...n=8...n=9...
0.|.1.....1.....1.....1.....1.....1.....1.....1.....1.......1.....
1.|.......1.....2.....3.....4.....5.....6.....7.....8.......9.....
2.|.............1.....3.....7....13....23....37....58......87.....
3.|...................1.....4....13....38...108...325....1275.....
4.|.........................1.....5....23...108...940..190214.....
5.|...............................1.....6....37...325..190214.....
6.|.....................................1.....7....58....1275.....
7.|...........................................1.....8......87.....
8.|.................................................1.......9.....
9.|.........................................................1.....
Sum.1.....2.....4.....8....17....38....98...306..1724..383172

W. M. B. Dukes, Tables of matroids.
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007 (see p. 7).
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
Index entries for sequences related to matroids

Row sums give A055545.
Columns include (truncated versions of) A000012 (k=0), A000027 (k=1), A058682 (k=2), A058693 (k=3).
Cf. A000041, A058669.

From Petros Hadjicostas, Oct 10 2019: (Start)
T(n,0) = 1 for n >= 0.
T(n,1) = n for n >= 1.
T(n,2) = -n + Sum_{k = 1..n} p(k) for n >= 2, where p(k) = A000041(k). [Dukes (2004), Theorem 2.1.] (End)

More terms from Jonathan Vos Post, Feb 14 2007

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar and Max Alekseyev

A058673 Number of matroids on n labeled points.

Original entry on oeis.org

1, 2, 5, 16, 68, 406, 3807, 75164, 10607540

Offset: 0

N. J. A. Sloane, Dec 30 2000

From Lorenzo Sauras Altuzarra, Aug 10 2023: (Start)
a(n) <= A014466(n).
a(n) <= A306020(n). (End)

			The 16 possible sets E such that ({1, 2, 3}, E) is a matroid:
  {{}}
  {{}, {1}}
  {{}, {2}}
  {{}, {3}}
  {{}, {1}, {2}}
  {{}, {1}, {3}}
  {{}, {2}, {3}}
  {{}, {1}, {2}, {3}}
  {{}, {1}, {2}, {1, 2}}
  {{}, {1}, {3}, {1, 3}}
  {{}, {2}, {3}, {2, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {2, 3}}
  {{}, {1}, {2}, {3}, {1, 3}, {2, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}
  {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

W. M. B. Dukes, Tables of matroids.
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
S. C. Locke, Matroids
Index entries for sequences related to matroids

Row sums of A058669. Closely related to A114491.

Cf. A014466 (abstract simplicial complexes), A055545 (unlabeled matroids), A306020.

A256157 Number of 2-polymatroids on n unlabeled points.

Original entry on oeis.org

1, 3, 10, 40, 228, 2380, 94495, 320863387

Offset: 0

Max Alekseyev, Mar 16 2015

Jayant Apte, JM Walsh, Constrained Linear Representability of Polymatroids and Algorithms for Computing Achievability Proofs in Network Coding, arXiv preprint arXiv:1605.04598, 2016
Thomas J. Savitsky, Enumeration of 2-polymatroids on up to seven elements. SIAM J. Discrete Math., 28(4):1641-1650, 2014. arXiv:1401.8006

Row sums of A256156.

Cf. A256158, A256159, A055545

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

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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A002773 Number of nonisomorphic simple matroids (or geometries) with n points.

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A053534 Triangle T(n,k) giving number of pairwise non-isomorphic (i.e., unlabeled) matroids of rank k on n points (n >= 0, 0 <= k <= n).

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A058673 Number of matroids on n labeled points.

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A256157 Number of 2-polymatroids on n unlabeled points.

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A174743 Partial sums of A076766.

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