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A350286
Number of different ways to partition the set of vertices of a convex (n+11)-gon into 4 nonintersecting polygons.
2
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
0,2
COMMENTS
Equivalently, the number of noncrossing set partitions of an (n+11)-set into 4 blocks with 3 or more elements in each block.
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
EXAMPLE
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.
MATHEMATICA
a[n_] := n*(n + 1)*(n + 2)*(n + 9)*(n + 10)*(n + 11)/144; Array[a, 35, 0] (* Amiram Eldar, Dec 26 2021 *)
CROSSREFS
Column k=4 of A350248.
Cf. A350116.
Sequence in context: A020182 A159746 A212408 * A250092 A330382 A140197
KEYWORD
easy,nonn
AUTHOR
Janaka Rodrigo, Dec 23 2021
STATUS
approved