A350286 Number of different ways to partition the set of vertices of a convex (n+11)-gon into 4 nonintersecting polygons.

0, 55, 286, 910, 2275, 4900, 9520, 17136, 29070, 47025, 73150, 110110, 161161, 230230, 322000, 442000, 596700, 793611, 1041390, 1349950, 1730575, 2196040, 2760736, 3440800, 4254250, 5221125, 6363630, 7706286, 9276085, 11102650, 13218400, 15658720, 18462136, 21670495, 25329150

Offset: 0

Author

Janaka Rodrigo, Dec 23 2021

Keywords

• easy
• nonn

Comments

Equivalently, the number of noncrossing set partitions of an (n+11)-set into 4 blocks with 3 or more elements in each block.

Examples

			a(1) = 55; solutions are {1,2,3} {4,5,6} {7,8,9} {10,11,12} with 3 different orientations, {1,2,3} {4,5,6} {11,12,7} {8,9,10} with 12 different orientations, {1,2,3} {12,4,5} {11,6,7} {8,9,10} with 12 different orientations, {1,2,3} {12,4,5} {10,11,6} {7,8,9} with 12 different orientations, {1,2,3} {4,5,6} {12,7,8} {9,10,11} with 12 orientations and {1,2,3} {4,8,12} {5,6,7} {9,10,11} with 4 orientations.
The above numbers can be considered to be the partition of a 12-set into 4 blocks or the partition of the vertices of a convex 12-gon into 4 triangles with vertices labeled 1,2,3,...,12 in order.
a(2) = 286 corresponding to the number of different ways to partition the vertices of a 13-gon into three triangles and one quadrilateral.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Column k=4 of A350248.

Cf. A350116.

Programs

Mathematica

a[n_] := n*(n + 1)*(n + 2)*(n + 9)*(n + 10)*(n + 11)/144; Array[a, 35, 0] (* Amiram Eldar, Dec 26 2021 *)

Formula

a(n) = (n*(n+1)*(n+2)*(n+9)*(n+10)*(n+11))/144.

G.f.: x*(55 - 99*x + 63*x^2 - 14*x^3)/(1 - x)^7. - Stefano Spezia, Dec 26 2021