A370772 -id:A370772 - 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).

A370771 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes rooted at a front.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 1, 9, 6, 2, 1, 1, 1, 20, 21, 6, 2, 1, 1, 1, 48, 83, 25, 6, 2, 1, 1, 1, 115, 356, 126, 25, 6, 2, 1, 1, 1, 286, 1599, 745, 135, 25, 6, 2, 1, 1, 1, 719, 7434, 4784, 895, 135, 25, 6, 2, 1, 1, 1, 1842, 35381, 32372, 6846, 915, 135, 25, 6, 2, 1, 1

Offset: 0

Andrew Howroyd, Mar 01 2024

A front is a k-clique.

			Triangle begins:
  1;
  1,   1;
  1,   1,    1;
  1,   2,    1,    1;
  1,   4,    2,    1,   1;
  1,   9,    6,    2,   1,   1;
  1,  20,   21,    6,   2,   1,  1;
  1,  48,   83,   25,   6,   2,  1, 1;
  1, 115,  356,  126,  25,   6,  2, 1, 1;
  1, 286, 1599,  745, 135,  25,  6, 2, 1, 1;
  1, 719, 7434, 4784, 895, 135, 25, 6, 2, 1, 1;
  ...

Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45.

Columns k=0..2 are A000012, A000081, A058866.

Cf. A370770 (unrooted), A370772, A370773.

A370773 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes rooted at a hedron with a designated front.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 2, 1, 0, 1, 12, 7, 2, 1, 0, 1, 30, 26, 7, 2, 1, 0, 1, 74, 110, 30, 7, 2, 1, 0, 1, 188, 481, 159, 30, 7, 2, 1, 0, 1, 478, 2209, 940, 168, 30, 7, 2, 1, 0, 1, 1235, 10379, 6093, 1104, 168, 30, 7, 2, 1, 0, 1, 3214, 49868, 41367, 8428, 1124, 168, 30, 7, 2, 1, 0

Offset: 0

Andrew Howroyd, Mar 01 2024

			Triangle begins:
  0;
  1,    0;
  1,    1,     0;
  1,    2,     1,    0;
  1,    5,     2,    1,    0;
  1,   12,     7,    2,    1,   0;
  1,   30,    26,    7,    2,   1,  0;
  1,   74,   110,   30,    7,   2,  1, 0;
  1,  188,   481,  159,   30,   7,  2, 1, 0;
  1,  478,  2209,  940,  168,  30,  7, 2, 1, 0;
  1, 1235, 10379, 6093, 1104, 168, 30, 7, 2, 1, 0;
  ...

Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45.

Columns k=1 is A000106.

Cf. A370770 (unrooted), A370771, A370772.

cp's OEIS Frontend

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

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A370771 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes rooted at a front.

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A370773 Triangle read by rows: T(n,k) is the number of k-trees with n unlabeled nodes rooted at a hedron with a designated front.

Views

Author

Keywords

Examples

Links

Crossrefs