A002055 Number of diagonal dissections of a convex n-gon into n-4 regions.

1, 9, 56, 300, 1485, 7007, 32032, 143208, 629850, 2735810, 11767536, 50220040, 212952285, 898198875, 3771484800, 15775723920, 65770848990, 273420862110, 1133802618000, 4691140763400, 19371432850770, 79850555673174

Offset: 5

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Number of standard tableaux of shape (n-4,n-4,1,1) (see Stanley reference). - Emeric Deutsch, May 20 2004

Number of increasing tableaux of shape (n-2,n-2) with largest entry 2n-6. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. - Oliver Pechenik, May 02 2014

a(n) = number of noncrossing partitions of 2n-6 into n-4 blocks, each of size at least 2. - Oliver Pechenik, May 02 2014

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 5..100
• D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
• A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
• P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
• O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, arXiv:1209.1355 [math.CO], 2012-2014.
• O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, 125 (2014), 357-378.
• R. C. Read, On general dissections of a polygon, Preprint (1974)
• Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388, Table 1.
• R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

Crossrefs

a(n) = f(n,n+1) where f is given in A034261.

Programs

Mathematica

Table[(Binomial[n-3,2]Binomial[2n-6,n-5])/(n-4),{n,5,30}] (* Harvey P. Dale, Nov 06 2011 *)

PARI

a(n) = (binomial(n - 3, 2) * binomial(2*n - 6, n - 5))/(n - 4);
for(n=5, 30, print1(a(n),", ")) \\ Indranil Ghosh, Apr 11 2017

Formula

a(n) = binomial(n-3, 2)*binomial(2*n-6, n-5)/(n-4).

With offset 0, this has a(n)=(n+2)*C(2n+4,n)/2 and e.g.f. dif(dif(x*dif(exp(2x)*Bessel_I(2,2x),x),x),x)/2. - Paul Barry, Aug 25 2007

G.f.: 16*x^5*(x+sqrt(1-4x))/((1-4x)^(3/2) *(1+sqrt(1-4x))^4 ). - R. J. Mathar, Nov 17 2011

D-finite with recurrence: (n-1)*a(n) +(23-11n)*a(n-1) +10*(4n-13)*a(n-2) +10*(23-5n)*a(n-3) +4*(2n-13)*a(n-4)=0. - R. J. Mathar, Nov 17 2011

a(n) ~ 4^n*sqrt(n)/(128*sqrt(Pi)). - Ilya Gutkovskiy, Apr 11 2017