A350248 -id:A350248 - cp's OEIS frontend

A350116 Number of ways to partition the set of vertices of a convex {n+8}-gon into 3 non-intersecting polygons.

0, 12, 45, 110, 220, 390, 637, 980, 1440, 2040, 2805, 3762, 4940, 6370, 8085, 10120, 12512, 15300, 18525, 22230, 26460, 31262, 36685, 42780, 49600, 57200, 65637, 74970, 85260, 96570, 108965, 122512, 137280, 153340, 170765, 189630, 210012, 231990, 255645, 281060, 308320

Offset: 0

Janaka Rodrigo, Dec 21 2021

Equivalently, the number of noncrossing set partitions of an {n+8}-set into 3 blocks with 3 or more elements in each block.

			The a(1) = 12 solutions are:
   {123}{456}{789}, {234}{567}{891}, {345}{678}{912},
   {156}{234}{567}, {267}{345}{891}, {378}{456}{912},
   {489}{567}{123}, {591}{678}{234}, {612}{789}{345},
   {723}{891}{456}, {834}{912}{567}, {945}{123}{678}.
In the above, the numbers can be considered to be the partition of a 9-set into 3 blocks or the partition of the vertices of a convex 9-gon into 3 triangles (with the vertices labeled 1..9 in order).
a(2) = 45 corresponding to the number of ways to partition the vertices of a 10-gon into two triangles and one quadrilateral.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Column k=3 of A350248.
The case of any number of parts for an n-gon is A114997.
The case of exactly 2 parts for a {n+5}-gon is A055998.

a[n_] := n*(n + 1)*(n + 7)*(n + 8)/12; Array[a, 40, 0] (* Amiram Eldar, Dec 21 2021 *)

a(n) = n*(n+1)*(n+7)*(n+8)/12.

G.f.: -x*(12-15*x+5*x^2)/(x-1)^5 . - R. J. Mathar, Aug 03 2022

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

Original entry on oeis.org

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

Janaka Rodrigo, Dec 23 2021

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

			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.

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

Column k=4 of A350248.

Cf. A350116.

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

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

A347862 Total number of polygons left out in all partitions of the set of vertices of a convex n-gon into nonintersecting polygons.

Original entry on oeis.org

0, 0, 0, 3, 7, 12, 39, 105, 231, 577, 1482, 3549, 8603, 21340, 52122, 126777, 310859, 761199, 1859014, 4549215, 11141085, 27266225, 66760855, 163567911, 400786617, 982265827, 2408361144, 5906499136, 14489105190, 35553445788, 87264949808, 214241203801

Offset: 3

Janaka Rodrigo, Jan 24 2022

nonn

			a(3) = a(4) = a(5) = 0 since the only partition of the vertices of a triangle, quadrilateral or pentagon into polygons is the full polygon so nothing is left out.
a(6) = 3 since the vertices of a hexagon can be partitioned into two non-intersecting triangles in A350248(6,2) = 3 ways and in each of these cases a quadrilateral is left over.
When partitioning the set of vertices of a convex 13-gon into 1 polygon, the number of polygons remaining is 0.
When partitioning it into 2 polygons, the remaining polygons are 52 quadrilaterals.
When partitioning it into 3 polygons, the remaining polygons are 65 hexagons + 650 quadrilaterals.
When partitioning it into 4 polygons, the remaining polygons are 13 octagons + 117 hexagons + 585 quadrilaterals.
This gives the total as 1482 polygons.

Partitioning into 3 polygons A350116.

Total number of different ways to partition the set of vertices of a convex polygon into nonintersecting polygons A350248.

seq(n)={my(p=O(x)); while(serprec(p,x)<=n, p = x + x*y*(1/(1 - x*p^2/(1 - p)) - 1)); Vec(subst(deriv(O(x*x^n) + p^3/(1-p), y), y, 1), 2-n) } \\ Andrew Howroyd, Jan 30 2022

More terms from Andrew Howroyd, Jan 30 2022

A350303 a(n) is the number of ways to partition the set of vertices of a convex (n+14)-gon into 5 nonintersecting polygons.

Original entry on oeis.org

0, 273, 1820, 7140, 21420, 54264, 122094, 251370, 482790, 876645, 1519518, 2532530, 4081350, 6388200, 9746100, 14535612, 21244356, 30489585, 43044120, 59865960, 82131896, 111275472, 149029650, 197474550, 259090650, 336817845, 434120778, 555060870, 704375490, 887564720, 1110986184

Offset: 0

Janaka Rodrigo, Dec 24 2021

Equivalently, the number of noncrossing set partitions of an (n+14)-set into 5 blocks with 3 or more elements in each block.

			The a(1)=273 solutions are {1,2,3} {4,5,6} {7,8,9} {10,11,12} {13,14,15} with its 3 different orientations and each of the following 18 patterns with its 15 orientations:
  {1,2,3} {4,5,15}  {6,7,8}   {9,10,11} {12,13,14}
  {1,2,3} {4,14,15} {5,6,7}   {8,9,10}  {11,12 13}
  {1,2,3} {4,5,6}   {7,8,15}  {9,10,11} {12 13,14}
  {1,2,3} {4,5,15}  {6,7,14}  {8,9,10}  {11,12,13}
  {1,2,3} {4,14,15} {5,12,13} {6,7,8}   {9,10,11}
  {1,2,3} {4,5,15}  {6,13,14} {7,8,9}   {10,11,12}
  {1,2,3} {4,14,15} {5,6,13}  {7,8,9}   {10,11,12}
  {1,2,3} {4,5,15}  {6,7,14}  {8,9,13}  {10,11,12}
  {1,2,3} {4,5,15}  {6,7,14}  {8,12,13} {9,10,11}
  {1,2,3} {4,5,15}  {6,13,14} {7,8,12}  {9,10,11}
  {1,2,3} {4,14,15} {5,12,13} {6,7,11}  {8,9,10}
  {1,2,3} {4,15,8}  {5,6,7}   {9,10,11} {12,13,14}
  {1,2,3} {4,15,8}  {5,6,7}   {9,13,14} {10,11,12}
  {1,2,3} {4,15,8}  {5,6,7}   {9,10,14} {11,12,13}
  {1,2,3} {4,5,15}  {6,7,8}   {9,10,14} {11,12,13}
  {1,2,3} {4,14,15} {5,6,7}   {8,12,13} {9,10,11}
  {1,2,3} {4,14,15} {5,6,7}   {8,9,13}  {10,11,12}
  {1,2,3} {4,5,15}  {6,7,8}   {9,13,14} {10,11,12}
In the above, the numbers can be considered to be the partition of a 15-set into 5 blocks or the partition of the vertices of a convex 15-gon into 5 triangles with vertices labeled 1,2,...,15 in order.
a(2)=1820 corresponding to the number of ways to partition the vertices of a 16-gon into 4 triangles and one quadrilateral.

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Column k=5 of A350248.

Cf. A350116.

a[n_] := n*(n + 1)*(n + 2)*(n + 3)*(n + 11)*(n + 12)*(n + 13)*(n + 14)/2880; Array[a, 30, 0] (* Amiram Eldar, Dec 26 2021 *)

a(n) = (1/2880)*n*(n+1)*(n+2)*(n+3)*(n+11)*(n+12)*(n+13)*(n+14).

G.f.: 7*x*(39 - 91*x + 84*x^2 - 36*x^3 + 6*x^4)/(1 - x)^9. - Stefano Spezia, Dec 26 2021

A350599 Number of ways to partition the set of vertices of a convex n-gon into non-intersecting directed polygons.

Original entry on oeis.org

2, 2, 2, 14, 30, 50, 170, 462, 1014, 2810, 7906, 19910, 53278, 148514, 397530, 1073918, 2976390, 8172426, 22413266, 62219830, 172846382, 479683762, 1338281802, 3743620974, 10475828630, 29389158426, 82643684034, 232644515366, 655928162878, 1852640651330, 5239096953274

Offset: 3

Janaka Rodrigo, Jan 08 2022

nonn

A directed polygon is a polygon with an associated direction (clockwise or counterclockwise).

Equivalently, the polygons can be colored using two colors. - Andrew Howroyd, Jan 09 2022

			a(7) = 2 + 28 = 30 since the 7-gon can be given two directions and the 7-gon can also be partitioned into a triangle and a quadrilateral in 7 different ways giving another 7 * 4 = 28 possibilities.

Cf. A350116, A350248.

a[n_] := Sum[2^k * Binomial[n + 1, k] * Binomial[n - 2*k - 1, k - 1]/(n + 1), {k, 1, Floor[n/3]}]; Array[a, 30, 3] (* Amiram Eldar, Jan 08 2022 *)

a(n) = sum(k=1, n\3, 2^k * binomial(n+1, k) * binomial(n-2*k-1, k-1)) / (n+1) \\ Andrew Howroyd, Jan 08 2022

a(n) = Sum_{k=1..floor(n/3)} 2^k * binomial(n+1, k) * binomial(n-2*k-1, k-1) / (n+1).
a(n) = Sum_{k=1..floor(n/3)} 2^k * A350248(n,k). - Andrew Howroyd, Jan 09 2022
The compositional inverse of x+Sum_{k=1..infinity} a_k x^{k+1} is x(1-x)/(1+x)(1-2x+x^2). Proved at MathOverflow 418996.

A352900 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into intersecting polygons.

Original entry on oeis.org

0, 0, 0, 7, 28, 79, 460, 2486, 11209, 59787, 361777, 2167635, 13577211, 91919186, 650059294, 4761980740, 36508824672, 292116858616, 2424047807182, 20847409357919, 185754041370693, 1711253802075941, 16272637412753211, 159561718074359537, 1611599794862761838, 16747401536440092104

Offset: 3

Janaka Rodrigo, Apr 07 2022

nonn

			For n=6, there are a(6) = 7 intersecting partitions of the convex hexagon. On vertices 1..6, they are the following pairs of triangles:
  {1,3,4}, {5,6,2}
  {4,5,1}, {2,3,6}
  {3,4,6}, {1,2,5}
  {2,3,5}, {1,4,6}
  {1,2,4}, {5,6,3}
  {1,6,3}, {5,4,2}
  {1,3,5}, {2,4,6}

Cf. A006505, A059022, A114997, A350248.

T2(n,k) = if (n<3, 0, if (k==1, 1, k*T2(n-1,k) + binomial(n-1,2)*T2(n-3,k-1))); \\ A059022
a5(n) = if (n<3, n==0, sum(k=1, n\3, T2(n,k))); \\ A006505
a7(n) = sum(k=ceil((n+3)/2), n, (1/(n+1) * binomial(n+1, k) * binomial(2*k-n-3, n-k)) ); \\ A114997
a(n) =  a5(n) - a7(n); \\ Michel Marcus, Apr 09 2022

a(n) = A006505(n) - A114997(n).
a(n) = Sum_{k=2..floor(n/3)} (T(n,k) - C(n+1,k)*C(n-2k-1,k-1)/(n+1)); n > 5, where T(n,k) = k*T(n-1,k) + C(n-1,2)*T(n-3,k-1); n > 5 and 1 < k <= floor(n/3), T(n,k) = 1 when k = 1.
T(n,k) = A059022(n,k) is the number of different ways to partition the set of vertices of a convex n-gon into k polygons.

More terms from Michel Marcus, Apr 09 2022

cp's OEIS Frontend

A350116 Number of ways to partition the set of vertices of a convex {n+8}-gon into 3 non-intersecting polygons.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A347862 Total number of polygons left out in all partitions of the set of vertices of a convex n-gon into nonintersecting polygons.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Extensions

A350303 a(n) is the number of ways to partition the set of vertices of a convex (n+14)-gon into 5 nonintersecting polygons.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A350599 Number of ways to partition the set of vertices of a convex n-gon into non-intersecting directed polygons.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A352900 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into intersecting polygons.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

Extensions