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A033282 Triangle read by rows: T(n,k) is the number of diagonal dissections of a convex n-gon into k+1 regions. 24
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

3,3

COMMENTS

T(n+3,k) is also the number of compatible k-sets 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, Mar 07 2003

Number of standard Young tableaux of shape (k+1,k+1,1^(n-k-3)), where 1^(n-k-3) denotes a sequence of n-k-3 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. - 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 f-vector of the simplicial complex dual to an associahedron of type A_n [Fomin & Reading, p.60]. See A001263 for the corresponding array of h-vectors for associahedra of type A_n. See A063007 and A080721 for the f-vectors for associahedra of type B and type D respectively. - Peter Bala, Oct 28 2008

f-vectors 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 Fulton-MacPherson compactification of the configuration space of n free particles on a line segment with a fixed particle at each end is the n-Dim Stasheff associahedron whose refined f-vector 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

S. Devadoss and J. O'Rourke, Discrete and Computational Geometry, Princeton Univ. Press, 2011 (See pg. 241)

G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.

G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 3..2000

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4. [From Tom Copeland, Nov 03 2008]

Paul Barry, On the Inverses of a Family of Pascal-Like 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), 256-257.

W. Butler, A. Kalotay and N. J. A. Sloane, Correspondence, 1974

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. (See p. 239.)

F. Chapoton, Enumerative properties of generalized associahedra

J. Cigler, Some remarks on lattice paths in strips along the x-axis, 2014.

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]

Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7

P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations, Discrete Math., 204, 1999, 203-229.

S. Fomin and N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004. [From Peter Bala, Oct 28 2008]

S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, arXiv:math/0104151 [math.RT], 2001.

S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) no.2, 497-529.

S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.

G. Kreweras, Sur les partitions non croisees d'un cycle, (French) Discrete Math. 1 (1972), no. 4, 333--350. MR0309747 (46 #8852)

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014

Vincent Pilaud, Brick polytopes, lattice quotients, and Hopf algebras, arXiv preprint, 2015.

R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.

R. Simion, Convex Polytopes and Enumeration, Adv. in Appl. Math. 18 (1997) pp. 149-180.

R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

R. Thomas, Lectures in Geometric Combinatorics [From Tom Copeland Oct 11 2011]

FORMULA

G.f. G=G(t, z) satisfies (1+t)*G^2 - z*(1-z-2*t*z)*G + t*z^4 = 0.

T(n, k) = binomial(n-3, k)*binomial(n+k-1, k)/(k+1) for n >= 3, 0 <= k <= n-3.

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[(1-x)^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 f-polynomials 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 h-polynomials 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(t-1),1/(t-1)]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/(1-x*y-(x+x*y)/(1-x*y/(1-(x+x*y)/(1-x*y/(1-(x+x*y)/(1-x*y/(1-.... (continued fraction). - Paul Barry, Feb 06 2009

Let h(t) = (1-t)^2/(1+(u-1)*(1-t)^2) = 1/(u + 2*t + 3*t^2 + 4*t^3 + ...), then a signed (n-1)-th row polynomial of A033282 is given by u^(2n-1)*(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[n-3, k]*Binomial[n+k-1, k]/(k+1);

Flatten[Table[t[n, k], {n, 3, 12}, {k, 0, n-3}]][[1 ;; 52]] (* Jean-François Alcover, Jun 16 2011 *)

CROSSREFS

Cf. 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 (f-vectors type B associahedra), A080721 (f-vectors type D associahedra), A126216 (mirror image). - Peter Bala, Oct 28 2008

Cf. A248727 for a relation to f-polynomials of simplices.

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

Missing factor of 2 for expansions of f1 and f2 added by Tom Copeland, Apr 12 2009

STATUS

approved

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Last modified September 5 03:49 EDT 2015. Contains 261339 sequences.