A319368 -id:A319368 - cp's OEIS frontend

A319367 Triangle read by rows: T(n,k) is the number of simple vertex transitive graphs with n nodes and valency k, (0 <= k < n).

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 0, 2, 0, 3, 0, 2, 0, 1, 1, 1, 2, 3, 4, 4, 3, 2, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 1, 4, 7, 11, 13, 13, 11, 7, 4, 1, 1, 1, 0, 1, 0, 3, 0, 4, 0, 3, 0, 1, 0, 1

Offset: 1

Andrew Howroyd, Sep 17 2018

			Triangle begins, n >= 1, 0 <= k < n:
  1;
  1, 1;
  1, 0, 1;
  1, 1, 1, 1;
  1, 0, 1, 0,  1;
  1, 1, 2, 2,  1,  1;
  1, 0, 1, 0,  1,  0,  1;
  1, 1, 2, 3,  3,  2,  1,  1;
  1, 0, 2, 0,  3,  0,  2,  0,  1;
  1, 1, 2, 3,  4,  4,  3,  2,  1,  1;
  1, 0, 1, 0,  2,  0,  2,  0,  1,  0,  1;
  1, 1, 4, 7, 11, 13, 13, 11,  7,  4,  1,  1;
  1, 0, 1, 0,  3,  0,  4,  0,  3,  0,  1,  0,  1;
  1, 1, 2, 3,  6,  6,  9,  9,  6,  6,  3,  2,  1,  1;
  1, 0, 3, 0,  8,  0, 12,  0, 12,  0,  8,  0,  3,  0, 1;
  1, 1, 3, 7, 16, 27, 40, 48, 48, 40, 27, 16,  7,  3, 1, 1;
  1, 0, 1, 0,  4,  0,  7,  0, 10,  0,  7,  0,  4,  0, 1, 0, 1;
  1, 1, 4, 7, 16, 24, 38, 45, 54, 54, 45, 38, 24, 16, 7, 4, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..496
B. D. McKay and G. F. Royle, The transitive graphs with at most 26 vertices, Ars Combin. 30 (1990), 161-176. (Annotated scanned copy)
Gordon Royle, Transitive Graphs.
Eric Weisstein's World of Mathematics, Vertex-Transitive Graph.

Columns k=2..12 (even n only for odd k) are A023645, A023646, A023647, A023640, A023641, A023642, A023643, A023644, A023637, A023638, A023639.
Row sums are A006799.
Cf. A319368, A319372.

A319372 Triangle read by rows: T(n,k) is the number of Cayley graphs with n nodes and valency k, (0 <= k < n).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 0, 2, 0, 3, 0, 2, 0, 1, 1, 1, 2, 2, 4, 4, 2, 2, 1, 1, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 1, 4, 7, 11, 13, 13, 11, 7, 4, 1, 1, 1, 0, 1, 0, 3, 0, 4, 0, 3, 0, 1, 0, 1

Offset: 1

Andrew Howroyd, Sep 17 2018

First differs from A319367 in row 10.

			Triangle begins, n >= 1, 0 <= k < n:
  1;
  1, 1;
  1, 0, 1;
  1, 1, 1, 1;
  1, 0, 1, 0,  1;
  1, 1, 2, 2,  1,  1;
  1, 0, 1, 0,  1,  0,  1;
  1, 1, 2, 3,  3,  2,  1,  1;
  1, 0, 2, 0,  3,  0,  2,  0,  1;
  1, 1, 2, 2,  4,  4,  2,  2,  1,  1;
  1, 0, 1, 0,  2,  0,  2,  0,  1,  0,  1;
  1, 1, 4, 7, 11, 13, 13, 11,  7,  4,  1,  1;
  1, 0, 1, 0,  3,  0,  4,  0,  3,  0,  1,  0,  1;
  1, 1, 2, 3,  6,  6,  9,  9,  6,  6,  3,  2,  1,  1;
  1, 0, 3, 0,  7,  0, 11,  0, 11,  0,  7,  0,  3,  0, 1;
  1, 1, 3, 7, 15, 26, 39, 47, 47, 39, 26, 15,  7,  3, 1, 1;
  1, 0, 1, 0,  4,  0,  7,  0, 10,  0,  7,  0,  4,  0, 1, 0, 1;
  1, 1, 4, 7, 16, 23, 38, 45, 53, 53, 45, 38, 23, 16, 7, 4, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..496
Gordon Royle, Transitive Graphs
Eric Weisstein's World of Mathematics, Cayley Graph
Wikipedia, Cayley graph

Column k=3 is aerated A319374.
Row sums are A185959.
Cf. A319367, A319368.

A006800 Number of connected vertex-transitive graphs with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 3, 10, 7, 18, 7, 64, 13, 51, 44, 272, 35, 365, 59, 1190, 235, 807, 187, 15422, 461, 4221, 1425, 25792, 1181, 46236, 2191, 677116, 6759, 132543, 11144, 1962756, 14601, 814155, 48447, 13102946, 52487, 9461929, 99879, 39133822, 399365, 34333611, 364723

Offset: 1

N. J. A. Sloane

B. D. McKay, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

B. D. McKay and G. F. Royle, The transitive graphs with at most 26 vertices, Ars Combin. 30 (1990), 161-176. (Annotated scanned copy)
B. D. McKay and G. F. Royle, The transitive graphs with at most 26 vertices, Ars Combin. 30 (1990), 161-176.
G. Royle, Transitive graphs
Steven Skiena, A Database of Graphs in Combinatorica Format.
Eric Weisstein's World of Mathematics, Vertex-Transitive Graph

Row sums of A319368.

Cf A006799.

A006799 = Cases[Import["https://oeis.org/A006799/b006799.txt", "Table"], {, }][[All, 2]];
Table[Sum[MoebiusMu[n/d] A006799[[d]], {d, Divisors[n]}], {n, 1, Length[A006799]}] (* Jean-François Alcover, Aug 20 2019 *)

Moebius transform of A006799. - Andrew Howroyd, Sep 18 2018

More terms from Vladeta Jovovic, Jun 30 2007

a(32)-a(47) from Andrew Howroyd, Nov 27 2018

cp's OEIS Frontend

A319367 Triangle read by rows: T(n,k) is the number of simple vertex transitive graphs with n nodes and valency k, (0 <= k < n).

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A319372 Triangle read by rows: T(n,k) is the number of Cayley graphs with n nodes and valency k, (0 <= k < n).

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A006800 Number of connected vertex-transitive graphs with n nodes.

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