

A033282


Triangle read by rows: T(n,k) is the number of diagonal dissections of a convex ngon into k+1 regions.


19



1, 1, 2, 1, 5, 5, 1, 9, 21, 14, 1, 14, 56, 84, 42, 1, 20, 120, 300, 330, 132, 1, 27, 225, 825, 1485, 1287, 429, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 1, 54, 936, 7644, 34398, 91728, 148512
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OFFSET

3,3


COMMENTS

T(n+3,k) is also the number of compatible ksets of cluster variables in Fomin and Zelevinsky's cluster algebra of finite type A_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of the triangle of Narayana numbers A001263. For example x^2+5*x+5=y^2+3*y+1.  Paul Boddington (psb(AT)maths.warwick.ac.uk), Mar 07 2003
Number of standard Young tableaux of shape (k+1,k+1,1^(nk3)), where 1^(nk3) denotes a sequence of nk3 1's (see the Stanley reference).
Number of k dimensional 'faces' of the n dimensional associahedron (see Simion, p. 168).  Mitch Harris, Jan 16 2007
Mirror image of triangle A126216 .  Philippe Deléham, Oct 19 2007
For relation to Lagrange inversion or series reversion and the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. [From Tom Copeland, Sep 29 2008]
Row generating polynomials 1/(n+1)*Jacobi_P(n,1,1,2*x+1). Row n of this triangle is the fvector of the simplicial complex dual to an associahedron of type A_n [Fomin & Reading, p.60]. See A001263 for the corresponding array of hvectors for associahedra of type A_n. See A063007 and A080721 for the fvectors for associahedra of type B and type D respectively. [From Peter Bala, Oct 28 2008]
fvectors of secondary polytopes for Grobner bases for optimization and integer programming (see De Loera et al. and Thomas).  Tom Copeland, Oct 11 2011
From Devadoss and O'Rourke's book: The FultonMacPherson compactification of the configuration space of n free particles on a line segment with a fixed particle at each end is the nDim Stasheff associahedron whose refined fvector is given in A133437 which reduces to A033282.  Tom Copeland, Nov 29 2011
Diagonals of A132081 are rows of A033282.  Tom Copeland, May 08 2012


REFERENCES

Paul Barry, On IntegerSequenceBased Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. [From Tom Copeland, Nov 03 2008]
P. Barry, On the Inverses of a Family of PascalLike Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256257.
A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237262 = Collected Mathematical Papers. Vols. 113, Cambridge Univ. Press, London, 18891897, Vol. 13, pp. 93ff. (See p. 239.)
S. Devadoss and J. O'Rourke, Discrete and Computational Geometry, Princeton Univ. Press, 2011 (See pg. 241)
B. Drake, I. M. Gessel and G. Xin, Three proofs and a generalization of the GouldenLitsynShevelev conjecture ..., J. Integer Sequences, Vol. 10 (2007), #07.3.7.
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math., 204, 1999, 203229.
S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) no.2, 497529.
S. Fomin and A. Zelevinsky, YSystems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 9771018.
Kreweras, G. Sur les partitions non croisees d'un cycle. (French) Discrete Math. 1 (1972), no. 4, 333350. MR0309747 (46 #8852)
R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370388.
R. Simion, "Convex Polytopes and Enumeration", Adv. in Appl. Math. 18 (1997) pp. 149180.
R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175177, 1996.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 3..2000
F. Chapoton, Enumerative properties of generalized associahedra
J. De Loera, J. Rambau, and F. Leal, Triangulations of Point Sets [From Tom Copeland Oct 11 2011]
S. Devadoss, Combinatorial Equivalence of Real Moduli Spaces [From Tom Copeland Nov 29 2011]
P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, Discrete Math., 204, 1999, 203229.
S. Fomin and N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/ParkCity 2004. [From Peter Bala, Oct 28 2008]
S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497529.
S. Fomin and A. Zelevinsky, Ysystems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 9771018.
R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370388.
R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175177, 1996.
R. Thomas, Lectures in Geometric Combinatorics [From Tom Copeland Oct 11 2011]


FORMULA

G.f. G=G(t, z) satisfies (1+t)G^2z(1z2tz)G+tz^4=0.
T(n, k)=binomial(n3, k)*binomial(n+k1, k)/(k+1) for n >= 3, 0 <=k <=n3.
Contribution from Tom Copeland, Nov 03 2008: (Start)
Two g.f.s (f1 and f2) for A033282 and their inverses (x1 and x2) can be derived from the Drake and Barry references.
1. a: f1(x,t) = y = {1  (2t+1) x  sqrt[1  (2t+1) 2x + x^2]}/[2x (t+1)]
= t x + (t + 2 t^2) x^2 + (t + 5 t^2 + 5 t^3) x^3 + ...
b: x1 = y/[t + (2t+1)y + (t+1)y^2] = y {1/[t/(t+1) + y]  1/(1+y)}
= (y/t)  (1+2t)(y/t)^2 + (1+ 3t + 3t^2)(y/t)^3 +...
2. a: f2(x,t) = y = {1  x  sqrt[(1x)^2  4xt]}/[2(t+1)]
= (t/(t+1)) x + t x^2 + (t + 2 t^2) x^3 + (t + 5 t^2 + 5 t^3) x^4 + ...
b: x2 = y(t+1) [1 y(t+1)]/[t + y(t+1)]
= (t+1) (y/t)  (t+1)^3 (y/t)^2 + (t+1)^4 (y/t)^3 + ...
c: y/x2(y,t) = [t/(t+1) + y] / [1 y(t+1)]
= t/(t+1) + (1+t) y + (1+t)^2 y^2 + (1+t)^3 y^3 + ...
x2(y,t) can be used along with the Lagrange inversion for an o.g.f. (A133437)
to generate A033282 and show that A133437 is a refinement of A033282,
i.e., a refinement of the fpolynomials of the associahedra, the Stasheff polytopes.
y/x2(y,t) can be used along with the indirect Lagrange inversion (A134264)
to generate A033282 and show that A134264 is a refinement of A001263, i.e.,
a refinement of the hpolynomials of the associahedra.
f1[x,t](t+1) gives a generator for A088617.
f1[xt,1/t](t+1) gives a generator for A060693, with inverse y/[1 + t + (2+t) y + y^2].
f1[x(t1),1/(t1)]t gives a generator for A001263, with inverse y/[t + (1+t) y + y^2].
The unsigned coefficients of x1(y t,t) are A074909, reverse rows of A135278. (End)
G.f.: 1/(1x*y(x+x*y)/(1x*y/(1(x+x*y)/(1x*y/(1(x+x*y)/(1x*y/(1.... (continued fraction). [From Paul Barry, Feb 06 2009]
Let h(t) = (1t)^2/(1+(u1)*(1t)^2) = 1/(u+2*t+3*t^2+4*t^3+...), then a signed (n1)th row polynomial of A033282 is given by u^(2n1)*(1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=2. The power series expansion of h(t) is related to A181289 (cf. A086810).  Tom Copeland, Sep 06 2011


EXAMPLE

Triangle begins:
1;
1,2;
1,5,5;
1,9,21,14;
1,14,56,84,42;


MATHEMATICA

t[n_, k_] = Binomial[n3, k]*Binomial[n+k1, k]/(k+1);
Flatten[Table[t[n, k], {n, 3, 12}, {k, 0, n3}]][[1 ;; 52]] (* JeanFrançois Alcover, Jun 16 2011 *)


CROSSREFS

Diagonals : A000012, A000096, A033275, A033276, A033277, A033278, A033279; A000108, A002054, A002055, A002056, A007160, A033280, A033281 Row sums : A001003 (Schroeder numbers, first term omitted) . See A086810 for another version.
A007160 is a diagonal. Cf. A001263.
With leading zero: A086810.
Cf. A019538 'faces' of the permutohedron.
Cf. A063007 (fvectors type B associahedra), A080721 (fvectors type D associahedra), A126216 (mirror image). [From Peter Bala, Oct 28 2008]
Sequence in context: A021468 A209830 A209695 * A126350 A204111 A079502
Adjacent sequences: A033279 A033280 A033281 * A033283 A033284 A033285


KEYWORD

nonn,tabl,easy


AUTHOR

N. J. A. Sloane.


EXTENSIONS

Added a missing factor of 2 for expansions of f1 and f2 Tom Copeland, Apr 12 2009


STATUS

approved



