Search: id:A033282
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%I A033282
%S A033282 1,1,2,1,5,5,1,9,21,14,1,14,56,84,42,1,20,120,300,330,132,1,27,225,
%T A033282 825,1485,1287,429,1,35,385,1925,5005,7007,5005,1430,1,44,616,4004,
%U A033282 14014,28028,32032,19448,4862,1,54,936,7644,34398,91728,148512
%N A033282 Triangle read by rows: T(n,k) is the number of diagonal dissections of 
               a convex n-gon into k+1 regions.
%C A033282 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 (psb(AT)maths.warwick.ac.uk), 
               Mar 07 2003
%C A033282 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).
%C A033282 Number of k dimensional 'faces' of the n dimensional associahedron (see 
               Simion, p. 168). - Mitch Harris (maharri(AT)gmail.com), Jan 16 2007
%C A033282 Mirror image of triangle A126216 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 19 2007
%C A033282 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 (tcjpn(AT)msn.com), Sep 29 2008]
%C A033282 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. 
               [From Peter Bala (pbala(AT)toucansurf.com), Oct 28 2008]
%D A033282 D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. 
               Monthly, 105 (1998), 256-257.
%D A033282 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.)
%D A033282 B. Drake, I. M. Gessel and G. Xin, Three proofs and a generalization 
               of the Goulden-Litsyn-Shevelev conjecture ..., J. Integer Sequences, 
               Vol. 10 (2007), #07.3.7.
%D A033282 P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, 
               Discrete Math., 204, 1999, 203-229.
%D A033282 S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, J. Amer. 
               Math. Soc. 15 (2002) no.2, 497-529.
%D A033282 S. Fomin and A. Zelevinsky, Y-Systems and generalized associahedra, Ann. 
               of Math. (2) 158 (2003), no. 3, 977-1018.
%D A033282 G. Kreweras, Sur les partitions..., Discrete Math. 1 (1972), 333-350.
%D A033282 R. C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 
               370-388.
%D A033282 R. Simion, "Convex Polytopes and Enumeration", Adv. in Appl. Math. 18 
               (1997) pp. 149-180.
%D A033282 R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. 
               Theory, Ser. A, 76, 175-177, 1996.
%D A033282 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 (tcjpn(AT)msn.com), Nov 03 2008]
%H A033282 F. Chapoton, <a href="http://www.mat.univie.ac.at/~slc/opapers/s51chapoton.html">
               Enumerative properties of generalized associahedra</a>
%H A033282 P. Flajolet and M. Noy, <a href="http://algo.inria.fr/flajolet/Publications/
               RR3196.ps.gz">Analytic Combinatorics of Non-crossing Configurations</
               a>, Discrete Math., 204, 1999, 203-229.
%H A033282 S. Fomin and A. Zelevinsky, <a href="http://arXiv.org/abs/math/0104151">
               Cluster algebras I: Foundations</a>, J. Amer. Math. Soc. 15 (2002), 
               no. 2, 497-529.
%H A033282 S. Fomin and A. Zelevinsky, <a href="http://projecteuclid.org/Dienst/
               Repository/1.0/Disseminate/euclid.annm/1080003770/body/pdf">Y-systems 
               and generalized associahedra</a>, Ann. of Math. (2) 158 (2003), no. 
               3, 977-1018.
%H A033282 R. C. Read, <a href="http://134.76.163.65/servlet/digbib?template=view.html&id=183269&startpage=376&endpage=3\
               94&image-path=http://134.76.176.141/cgi-bin/letgifsfly.cgi&image-subpath=/
               4789&image-subpath=4789&pagenumber=376&imageset-id=4789">On general 
               dissections of a polygon</a>, Aequat. Math. 18 (1978), 370-388.
%H A033282 R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/poldis.ps">
               Polygon dissections and standard Young tableaux</a>, J. Comb. Theory, 
               Ser. A, 76, 175-177, 1996.
%H A033282 S. Fomin, N. Reading, <a href="http://arxiv.org/abs/math.CO/0505518">
               Root systems and generalized associahedra</a>, Lecture notes for 
               IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct 
               28 2008]
%F A033282 G.f. G=G(t, z) satisfies (1+t)G^2-z(1-z-2tz)G+tz^4=0.
%F A033282 T(n, k)=binomial(n-3, k)*binomial(n+k-1, k)/(k+1) for n >= 3, 0 <=k <=n-3.
%F A033282 Contribution from Tom Copeland (tcjpn(AT)msn.com), Nov 03 2008: (Start)
%F A033282 Two g.f.s (f1 and f2) for A033282 and their inverses (x1 and x2) can 
               be derived from the Drake and Barry references.
%F A033282 1. a: f1(x,t) = y = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/[2x (t+1)]
%F A033282 = t x + (t + 2 t^2) x^2 + (t + 5 t^2 + 5 t^3) x^3 + ...
%F A033282 b: x1 = y/[t + (2t+1)y + (t+1)y^2] = y {1/[t/(t+1) + y] - 1/(1+y)}
%F A033282 = (y/t) - (1+2t)(y/t)^2 + (1+ 3t + 3t^2)(y/t)^3 +...
%F A033282 2. a: f2(x,t) = y = {1 - x - sqrt[(1-x)^2 - 4xt]}/[2(t+1)]
%F A033282 = (t/(t+1)) x + t x^2 + (t + 2 t^2) x^3 + (t + 5 t^2 + 5 t^3) x^4 + ...
%F A033282 b: x2 = y(t+1) [1- y(t+1)]/[t + y(t+1)]
%F A033282 = (t+1) (y/t) - (t+1)^3 (y/t)^2 + (t+1)^4 (y/t)^3 + ...
%F A033282 c: y/x2(y,t) = [t/(t+1) + y] / [1- y(t+1)]
%F A033282 = t/(t+1) + (1+t) y + (1+t)^2 y^2 + (1+t)^3 y^3 + ...
%F A033282 x2(y,t) can be used along with the Lagrange inversion for an o.g.f. (A133437)
%F A033282 to generate A033282 and show that A133437 is a refinement of A033282,
%F A033282 i.e., a refinement of the f-polynomials of the associahedra, the Stasheff 
               polytopes.
%F A033282 y/x2(y,t) can be used along with the indirect Lagrange inversion (A134264)
%F A033282 to generate A033282 and show that A134264 is a refinement of A001263, 
               i.e.,
%F A033282 a refinement of the h-polynomials of the associahedra.
%F A033282 f1[x,t](t+1) gives a generator for A088617.
%F A033282 f1[xt,1/t](t+1) gives a generator for A060693, with inverse y/[1 + t 
               + (2+t) y + y^2].
%F A033282 f1[x(t-1),1/(t-1)]t gives a generator for A001263, with inverse y/[t 
               + (1+t) y + y^2].
%F A033282 The unsigned coefficients of x1(y t,t) are A074909, reverse rows of A135278. 
               (End)
%F A033282 G.f.: 1/(1-xy-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-.... (continued 
               fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 06 2009]
%e A033282 1; 1,2; 1,5,5; 1,9,21,14; 1,14,56,84,42;
%Y A033282 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.
%Y A033282 A007160 is a diagonal. Cf. A001263.
%Y A033282 With leading zero: A086810.
%Y A033282 Cf. A019538 'faces' of the permutohedron.
%Y A033282 Cf. A063007 (f-vectors type B associahedra), A080721 (f-vectors type 
               D associahedra), A126216 (mirror image). [From Peter Bala (pbala(AT)toucansurf.com), 
               Oct 28 2008]
%Y A033282 Sequence in context: A145882 A111785 A021468 this_sequence A126350 A079502 
               A126124
%Y A033282 Adjacent sequences: A033279 A033280 A033281 this_sequence A033283 A033284 
               A033285
%K A033282 nonn,tabl,easy
%O A033282 3,3
%A A033282 N. J. A. Sloane (njas(AT)research.att.com).
%E A033282 Added a missing factor of 2 for expansions of f1 and f2 Tom Copeland 
               (tcjpn(AT)msn.com), Apr 12 2009

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