A380361 - OEIS

%I #12 Jan 26 2025 17:42:42

%S 1,0,1,0,1,1,0,0,1,1,0,0,1,2,1,0,0,0,4,2,1,0,0,0,4,8,3,1,0,0,0,0,12,

%T 16,3,1,0,0,0,0,6,40,25,4,1,0,0,0,0,0,43,93,40,4,1,0,0,0,0,0,19,165,

%U 197,56,5,1,0,0,0,0,0,0,143,505,364,80,5,1

%N Triangle read by rows: T(n,k) is the number of embeddings on the sphere of Halin graphs on n unlabeled nodes with circuit rank k up to orientation-preserving homeomorphisms, 3 <= k <= n-1.

%C 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.

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

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

%H Andrew Howroyd, <a href="/A380361/b380361.txt">Table of n, a(n) for n = 4..1278</a> (first 50 rows)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HalinGraph.html">Halin Graph</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Circuit_rank">Circuit rank</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Halin_graph">Halin graph</a>.

%F T(n,k) = A295633(k, n-k).

%e Triangle begins:

%e   n\k| 3  4  5  6   7   8    9   10  11  12  13

%e -----+-----------------------------------------

%e    4 | 1;

%e    5 | 0, 1;

%e    6 | 0, 1, 1;

%e    7 | 0, 0, 1, 1;

%e    8 | 0, 0, 1, 2,  1;

%e    9 | 0, 0, 0, 4,  2,  1;

%e   10 | 0, 0, 0, 4,  8,  3,   1;

%e   11 | 0, 0, 0, 0, 12, 16,   3,   1;

%e   12 | 0, 0, 0, 0,  6, 40,  25,   4,  1;

%e   13 | 0, 0, 0, 0,  0, 43,  93,  40,  4,  1;

%e   14 | 0, 0, 0, 0,  0, 19, 165, 197, 56,  5,  1;

%e   ...

%o (PARI) \\ See PARI Link in A380362 for program code.

%o { my(T=A380361rows(12)); for(i=1, #T, print(T[i])) }

%Y Row sums are A380360.

%Y Column sums are A003455.

%Y Main diagonal is A000012.

%Y Central coefficients are A001683.

%Y Cf. A295633, A380362.

%K nonn,tabl

%O 4,14

%A _Andrew Howroyd_, Jan 25 2025