A202037 -id:A202037 - cp's OEIS frontend

A370770 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 6, 5, 2, 1, 1, 1, 1, 11, 12, 5, 2, 1, 1, 1, 1, 23, 39, 15, 5, 2, 1, 1, 1, 1, 47, 136, 58, 15, 5, 2, 1, 1, 1, 1, 106, 529, 275, 64, 15, 5, 2, 1, 1, 1, 1, 235, 2171, 1505, 331, 64, 15, 5, 2, 1, 1, 1

Offset: 0

Andrew Howroyd, Mar 01 2024

			Triangle begins:
  1;
  1,   1;
  1,   1,   1;
  1,   1,   1,   1;
  1,   2,   1,   1,  1;
  1,   3,   2,   1,  1,  1;
  1,   6,   5,   2,  1,  1, 1;
  1,  11,  12,   5,  2,  1, 1, 1;
  1,  23,  39,  15,  5,  2, 1, 1, 1;
  1,  47, 136,  58, 15,  5, 2, 1, 1, 1;
  1, 106, 529, 275, 64, 15, 5, 2, 1, 1, 1;
  ...

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45.
I. M. Gessel and A. Gainer-Dewar, Counting unlabeled k-trees, arXiv:1309.1429 [math.CO], 2013-2014.
I. M. Gessel and A. Gainer-Dewar, Counting unlabeled k-trees, J. Combin. Theory Ser. A 126 (2014), 177-193.
Andrew Howroyd, SageMath Program code (from Andrew Gainer-Dewar reference).

Columns k=0..7 are A000012, A000055, A054581, A078792, A078793, A201702, A202037, A322754.

Cf. A135021 (labeled version), A370771, A370772, A370773.

T(n,k) = A370771(n,k) + A370772(n,k) - A370773(n,k).

A322754 Number of unlabeled 7-trees on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 5, 15, 64, 342, 2344, 19137, 181098, 1922215, 22472875, 284556458, 3849828695, 54974808527, 819865209740, 12655913153775, 200748351368185, 3253193955012557, 53619437319817482, 895778170144927928, 15129118461773051724

Offset: 1

N. J. A. Sloane, Dec 26 2018

nonn

A k-tree is recursively defined as follows: K_k is a k-tree and any k-tree on n+1 vertices is obtained by joining a vertex to a k-clique in a k-tree on n vertices.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 328.

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45.

Column k=7 of A370770.

Cf. A054581 (unlabeled 2-trees), A078792 (unlabeled 3-trees), A078793 (unlabeled 4-trees), A201702 (unlabeled 5-trees), A202037 (unlabeled 6-trees).

A224917 Stable k-tree numbers.

Original entry on oeis.org

1, 1, 1, 2, 5, 15, 64, 342, 2344, 19137, 181204, 1927017, 22652805, 290392448, 4022276630, 59749492128, 946174967813, 15892939156209

Offset: 0

Ira M. Gessel, Apr 19 2013

a(n) is the number of unlabeled k-trees with n+k vertices for all k >= n-2.

A k-tree is recursively defined as follows: The complete graph K_k is a k-tree and a k-tree on n+1 vertices is obtained by joining a vertex to a k-clique in a k-tree on n vertices.

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
A. Gainer-Dewar, Γ-species and the enumeration of k-trees, Electron. J. Combin. 19, no. 4, (2012), P45.
I. M. Gessel and A. Gainer-Dewar, Counting unlabeled k-trees, arXiv:1309.1429 [math.CO], 2013-2014.
I. M. Gessel and A. Gainer-Dewar, Counting unlabeled k-trees, J. Combin. Theory Ser. A 126 (2014), 177-193.

Cf. A000055 (unlabeled trees), A054581 (unlabeled 2-trees), A078792 (unlabeled 3-trees), A078793 (unlabeled 4-trees), A201702 (unlabeled 5-trees), A202037 (unlabeled 6-trees).

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A370770 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes.

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A322754 Number of unlabeled 7-trees on n nodes.

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A224917 Stable k-tree numbers.

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