A380362 Triangle read by rows: T(n,k) is the number of Halin graphs on n unlabeled nodes with circuit rank k, 3 <= k <= n-1.

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 3, 2, 1, 0, 0, 0, 3, 6, 3, 1, 0, 0, 0, 0, 7, 11, 3, 1, 0, 0, 0, 0, 4, 24, 17, 4, 1, 0, 0, 0, 0, 0, 24, 51, 26, 4, 1, 0, 0, 0, 0, 0, 12, 89, 109, 36, 5, 1, 0, 0, 0, 0, 0, 0, 74, 265, 194, 50, 5, 1, 0, 0, 0, 0, 0, 0, 27, 371, 660, 345, 65, 6, 1

Offset

4,14

Comments

The circuit rank is equal to the number of leaves on the tree before it is extended into a Halin graph by joining up the leaves.

The main diagonal of the graph corresponds with the wheel graphs which have the greatest circuit rank of all Halin graphs.

T(n,k) is also the number of nonequivalent dissections of a k-gon into n-k polygons by nonintersecting diagonals up to rotations and reflections.

Links

Andrew Howroyd, Table of n, a(n) for n = 4..1278 (first 50 rows)

Andrew Howroyd, Formulas and PARI Program, Jan 2025.

Eric Weisstein's World of Mathematics, Halin Graph.

Wikipedia, Circuit rank.

Wikipedia, Halin graph.

Formula

T(n,k) = A295634(k, n-k).

Example

Triangle begins:
  n\k| 3  4  5  6  7   8   9   10  11  12  13
-----+----------------------------------------
   4 | 1;
   5 | 0, 1;
   6 | 0, 1, 1;
   7 | 0, 0, 1, 1;
   8 | 0, 0, 1, 2, 1;
   9 | 0, 0, 0, 3, 2,  1;
  10 | 0, 0, 0, 3, 6,  3,  1;
  11 | 0, 0, 0, 0, 7, 11,  3,   1;
  12 | 0, 0, 0, 0, 4, 24, 17,   4,  1;
  13 | 0, 0, 0, 0, 0, 24, 51,  26,  4,  1;
  14 | 0, 0, 0, 0, 0, 12, 89, 109, 36,  5,  1;
   ...

Prog

(PARI) \\ See PARI Link for program code.
{ my(T=A380361rows(12)); for(i=1, #T, print(T[i])) }

Crossrefs

Row sums are A346779.

Column sums are A001004.

Main diagonal is A000012.

Central coefficients are A000207.

Cf. A295634, A380361.

Sequence in context: A103294 A198379 A079680 * A037855 A037873 A036869

Adjacent sequences: A380359 A380360 A380361 * A380363 A380364 A380365

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 25 2025

Status

approved