A002056 Number of diagonal dissections of a convex n-gon into n-5 regions.

1, 14, 120, 825, 5005, 28028, 148512, 755820, 3730650, 17978180, 84987760, 395482815, 1816357725, 8250123000, 37119350400, 165645101160, 733919156190, 3231337461300, 14147884842000, 61636377252450, 267325773340626, 1154761882042824, 4969989654817600

Offset: 6

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of standard tableaux of shape (n-5,n-5,1,1,1) (see Stanley reference). - Emeric Deutsch, May 20 2004

Number of increasing tableaux of shape (n-2,n-2) with largest entry 2n-7. 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

Number of noncrossing partitions of 2n-7 into n-5 blocks all 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=6..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.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série, FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics; arXiv:0912.0072 [math.NT], 2009.
• 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.

Programs

Magma

[Binomial(n-3, 3)*Binomial(2*n-7, n-6)/(n-5): n in [6..30]]; // Vincenzo Librandi, Feb 18 2020

Maple

A002056:=n->binomial(n-3,3)*binomial(2*n-7,n-6)/(n-5): seq(A002056(n), n=6..40); # Wesley Ivan Hurt, Apr 12 2017

Mathematica

Table[Binomial[n - 3, 3] Binomial[2n - 7, n - 6]/(n - 5), {n, 6, 50}] (* Indranil Ghosh, Apr 11 2017 *)

Formula

a(n) = binomial(n-3, 3)*binomial(2n-7, n-6)/(n-5).

G.f.: (x-1+(1-11*x+40*x^2-50*x^3+10*x^4)*(1-4*x)^(-5/2))/(2*x^5). - Mark van Hoeij, Oct 25 2011

a(n) ~ 4^n*n^(3/2)/(768*sqrt(Pi)). - Ilya Gutkovskiy, Apr 11 2017

D-finite with recurrence: -(n-1)*(n-5)*(n-6)*a(n) +2*(2*n-7)*(n-3)*(n-4)*a(n-1)=0. - R. J. Mathar, Feb 16 2020