|
| |
|
|
A295260
|
|
Array read by antidiagonals: T(n,k) = number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals up to rotation and reflection (k >= 3).
|
|
16
|
|
|
|
1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 5, 4, 1, 1, 3, 8, 16, 12, 1, 1, 3, 12, 33, 60, 27, 1, 1, 4, 16, 68, 194, 261, 82, 1, 1, 4, 21, 112, 483, 1196, 1243, 228, 1, 1, 5, 27, 183, 1020, 3946, 8196, 6257, 733, 1, 1, 5, 33, 266, 1918, 10222, 34485, 58140, 32721, 2282
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,9
|
|
|
COMMENTS
|
The polygon prior to dissection will have n*(k-2)+2 sides.
In the Harary, Palmer and Read reference these are the sequences called h.
T(n,k) is the number of unoriented polyominoes containing n k-gonal tiles of the hyperbolic regular tiling with Schläfli symbol {k,oo}. Stereographic projections of several of these tilings on the Poincaré disk can be obtained via the Christensson link. For unoriented polyominoes, chiral pairs are counted as one. T(n,2) could represent polyominoes of the Euclidean regular tiling with Schläfli symbol {2,oo}; T(n,2) = 1. - Robert A. Russell, Jan 21 2024
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n,k) ~ A295222(n,k)/(2*n) for fixed k.
|
|
|
EXAMPLE
|
Array begins:
===================================================
n\k| 3 4 5 6 7 8
---|-----------------------------------------------
1 | 1 1 1 1 1 1 ...
2 | 1 1 1 1 1 1 ...
3 | 1 2 2 3 3 4 ...
4 | 3 5 8 12 16 21 ...
5 | 4 16 33 68 112 183 ...
6 | 12 60 194 483 1020 1918 ...
7 | 27 261 1196 3946 10222 22908 ...
8 | 82 1243 8196 34485 109947 290511 ...
9 | 228 6257 58140 315810 1230840 3844688 ...
10 | 733 32721 427975 2984570 14218671 52454248 ...
...
|
|
|
MATHEMATICA
|
u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
T[n_, k_] := (u[n, k, 1] + If[OddQ[n], u[(n - 1)/2, k, Quotient[k, 2]], If[OddQ[k], (u[n/2 - 1, k, k - 1] + u[n/2, k, 1])/2, u[n/2, k, 1]]] + (If[EvenQ[n], u[n/2, k, 1]] - u[n, k, 2])/2 + DivisorSum[GCD[n - 1, k], EulerPhi[#]*u[(n - 1)/#, k, k/#] &]/k)/2 /. Null -> 0;
Table[T[n - k + 2, k + 1], {n, 1, 11}, {k, n + 1, 2, -1}] // Flatten (* Jean-François Alcover, Dec 28 2017, after Andrew Howroyd *)
|
|
|
PROG
|
(PARI) \\ here u is Fuss-Catalan sequence with p = k+1.
u(n, k, r) = {r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n, k) = {(u(n, k, 1) + if(n%2, u((n-1)/2, k, k\2), if(k%2, (u(n/2-1, k, (k-1)) + u(n/2, k, 1))/2, u(n/2, k, 1))) + (if(n%2==0, u(n/2, k, 1))-u(n, k, 2))/2 + sumdiv(gcd(n-1, k), d, eulerphi(d)*u((n-1)/d, k, k/d))/k)/2}
for(n=1, 10, for(k=3, 8, print1(T(n, k), ", ")); print);
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
