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

Author

Andrew Howroyd, Jan 25 2025

Keywords

• nonn
• tabl

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.

Examples

			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;
   ...
		

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.

Crossrefs

Row sums are A346779.

Column sums are A001004.

Main diagonal is A000012.

Central coefficients are A000207.

Cf. A295634, A380361.

Programs

PARI

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

Formula

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