A350286 -id:A350286 - cp's OEIS frontend

A350248 Triangle read by rows: T(n,k) is the number of noncrossing partitions of an n-set into k blocks of size 3 or more, n >= 0, 0 <= k <= floor(n/3).

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 3, 0, 1, 7, 0, 1, 12, 0, 1, 18, 12, 0, 1, 25, 45, 0, 1, 33, 110, 0, 1, 42, 220, 55, 0, 1, 52, 390, 286, 0, 1, 63, 637, 910, 0, 1, 75, 980, 2275, 273, 0, 1, 88, 1440, 4900, 1820, 0, 1, 102, 2040, 9520, 7140, 0, 1, 117, 2805, 17136, 21420, 1428

Offset: 0

Andrew Howroyd and Janaka Rodrigo, Dec 21 2021

			Triangle begins:
  1;
  0;
  0;
  0, 1;
  0, 1;
  0, 1;
  0, 1,   3;
  0, 1,   7;
  0, 1,  12;
  0, 1,  18,   12;
  0, 1,  25,   45;
  0, 1,  33,  110;
  0, 1,  42,  220,   55;
  0, 1,  52,  390,  286;
  0, 1,  63,  637,  910;
  0, 1,  75,  980, 2275,  273;
  0, 1,  88, 1440, 4900, 1820;
  0, 1, 102, 2040, 9520, 7140;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1750 (rows 0..100)

Columns k=2..5 are A055998, A350116, A350286, A350303.
Row sums are A114997.
Cf. A001263 (blocks of any size), A108263 (blocks of size 2 or more).

T(n)={my(p=1+O(x^3)); for(i=1, n\3, p=1+y*(x*p)^3/(1-x*p)); [Vecrev(t)| t<-Vec(p + O(x*x^n))]}
{my(A=T(12)); for(i=1, #A, print(A[i]))}

T(n,k) = if(n==0 || k>n\3, k==0, binomial(n+1, n-k+1) * binomial(n-2*k-1, k-1) / (n+1)) \\ Andrew Howroyd, Dec 31 2021

G.f.: A(x,y) satisfies A(x,y) = 1 + y*(x*A(x,y))^3/(1 - x*A(x,y)).

T(n,k) = binomial(n+1, n-k+1) * binomial(n-2*k-1, k-1) / (n+1) for n > 0.

A352477 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into 4 intersecting polygons.

Original entry on oeis.org

15345, 199914, 1610700, 10333050, 57958005, 297787980, 1439757200, 6662364668, 29844152321, 130445708134, 559533869356, 2365296230374, 9885290683829, 40944327268760, 168389163026240, 688631375953560, 280357073972529, 11373212442818370, 46006062638648940

Offset: 12

Janaka Rodrigo, Mar 17 2022

nonn

			The set of vertices of a convex 14-gon can be partitioned into 4 polygons in 1611610 different ways:
- 3 triangles and 1 pentagon in (1/3!)*C(14,3)*C(11,3)*C(8,3)*C(5,5) = 560560 different ways, and
- 2 triangles and 2 quadrilaterals in (1/2!)*(1/2!)*C(14,3)*C(11,3)*C(8,4)*C(4,4) = 1051050 ways.
Subtracting the A350286(14-11)=910 nonintersecting partitions leaves a(14)=1610700.

Cf. A261724, A350116, A350286.

a4(n) = (1/12)*(-3^(n - 2)*(n^2 + 5*n + 18) + (1/64)*(2^(2*n + 5) + 3*2^n*(n^4 + 2*n^3 + 19*n^2 + 42*n + 64) - 16*(n^6 - 9*n^5 + 43*n^4 - 91*n^3 + 112*n^2 - 32*n + 8))); \\ A261724
a6(n) = (n*(n+1)*(n+2)*(n+9)*(n+10)*(n+11))/144; \\ A350286
a(n) = a4(n) - a6(n-11); \\ Michel Marcus, Mar 20 2022

a(n) = A261724(n) - A350286(n - 11), n > 11.

cp's OEIS Frontend

A350248 Triangle read by rows: T(n,k) is the number of noncrossing partitions of an n-set into k blocks of size 3 or more, n >= 0, 0 <= k <= floor(n/3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A352477 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into 4 intersecting polygons.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula