Greg Kuperberg - cp's OEIS frontend

User: Greg Kuperberg

Greg Kuperberg has authored 3 sequences.

A060050 Number of irreducible nonpositively curved triangulations of an n-gon: All internal vertices have at valence at least 6 and no diagonals of the n-gon are allowed.

0, 1, 0, 0, 1, 1, 5, 13, 46, 155, 561, 2068, 7871, 30586, 121391, 490196, 2011422, 8370698, 35285987, 150485667, 648653910, 2823402675, 12400659846, 54920758496, 245126368841, 1101983749921, 4987538210079, 22716326086134

Offset: 2

Greg Kuperberg, Feb 15 2001

			c(8) = 5 = 1+4. We can divide the octagon into 8 pie slices and we can split any pair of opposite radii of this triangulation into two triangles.

G. Kuperberg, Spiders for rank 2 Lie algebras, arXiv:q-alg/9712003, 1997.
G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), 109-151.

Cf. A060049, A059710.

The g.f. C(x) is derived from the g.f. B(x) of A060049 by B_1(x) = C_1(B_1(x))+x, where B_1(x) = B(x)/x and C_1(x) = C(x)/x.

A060049 Triangulations of an n-gon such that each internal vertex has valence at least 6, i.e., nonpositively curved triangulations.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 15, 50, 181, 697, 2821, 11892, 51874, 232974, 1073070, 5053029, 24264565, 118570292, 588567257, 2963358162, 15114174106, 78004013763, 406971280545, 2144659072330, 11407141925639, 61197287846831

Offset: 0

Greg Kuperberg, Feb 15 2001

This is the connected version of A059710 in the following sense. Let C(x) be the ordinary generating function for this sequence and A(x) the ordinary generating function for A059710. Then these satisfy the functional equation A(x) = C(x*A(x)). - Bruce Westbury, Nov 05 2013

			a(6) = 15 because there are 14 = A000108(4) triangulations without internal vertices, plus the triangulation with 6 pie slices.

Bruce Westbury, Table of n, a(n) for n = 0..39
Greg Kuperberg, Spiders for rank 2 Lie algebras, arXiv:q-alg/9712003, 1997.
Greg Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), 109-151.
Bruce W. Westbury, Enumeration of non-positive planar trivalent graphs, arXiv:math/0507112 [math.CO], 2005.
Bruce W. Westbury, Enumeration of non-positive planar trivalent graphs, J. Algebraic Combin. 25 (2007)

Cf. A059710.

The g.f. B(x) is derived from the g.f. A(x) of A059710 by A(x) = A(x*B(x))+1.

A059710 Dimension of space of invariants of n-th tensor power of 7-dimensional irreducible representation of G_2. Also the number of n-leaf, otherwise trivalent graphs in a disk such that all faces have at least 6 sides.

Original entry on oeis.org

1, 0, 1, 1, 4, 10, 35, 120, 455, 1792, 7413, 31780, 140833, 641928, 3000361, 14338702, 69902535, 346939792, 1750071307, 8958993507, 46484716684, 244187539270, 1297395375129, 6965930587924, 37766629518625

Offset: 0

Greg Kuperberg, Feb 08 2001

Related to triangulations of an n-gon such that all internal vertices have valence at least 6.

This sequence arises from the sequence G_2 polynomials in q when q is replaced by 1. The sequence of degrees of these q-polynomials (Westbury 2010) is A227849. - Michael Somos, Nov 01 2013

			G.f. = 1 + x^2 + x^3 + 4*x^4 + 10*x^5 + 35*x^6 + 120*x^7 + 455*x^8 + ...

Alec Mihailovs, A Combinatorial Approach to Representations of Lie Groups and Algebras, Birkhäuser Boston (2003).

Michael De Vlieger, Table of n, a(n) for n = 0..1200
Georgia Benkart and A. Elduque, Cross products, invariants, and centralizers, arXiv preprint arXiv:1606.07588 [math.RT], 2016.
Alin Bostan, Jordan Tirrell, Bruce W. Westbury, and Yi Zhang, On sequences associated to the invariant theory of rank two simple Lie algebras, arXiv:1911.10288 [math.CO], 2019.
Alin Bostan, Jordan Tirrell, Bruce W. Westbury, and Yi Zhang, On some combinatorial sequences associated to invariant theory, arXiv:2110.13753 [math.CO], 2021.
Juan B. Gil and Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018. Also Discrete Mathematics (2019) Article 111705. doi:10.1016/j.disc.2019.111705
G. Kuperberg, Spiders for rank 2 Lie algebras, arXiv:q-alg/9712003, 1997.
G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), 109-151.
Gilles Lachaud, The distribution of the trace in the compact group of type G_2, in Arithmetic Geometry: Contemporary Mathematics (2019) Vol. 722, 79-103.
Q. Lu, W. Zheng, and Z. Zheng, On the distribution of Jacobi sums, arXiv:1305.3405 [math.NT], 2013.
Robert Scherer, A criterion for asymptotic sharpness in the enumeration of simply generated trees, arXiv:2003.07984 [math.CO], 2020.
Robert Scherer, Topics in Number Theory and Combinatorics, Ph. D. Dissertation, Univ. of California Davis (2021).
Bruce W. Westbury, Enumeration of non-positive planar trivalent graphs, arXiv:math/0507112 [math.CO], 2005.
Bruce W. Westbury, Enumeration of non-positive planar trivalent graphs, J. Algebraic Combin. 25 (2007).
Bruce W. Westbury, Finding recurrence relation for a sequence of polynomials (2010).
Bruce W. Westbury, Invariant tensors and the cyclic sieving phenomenon, El. J. Combinat. 23 (4) (2016) P4.2

The analogous sequence for A_1 is A000108.

See A060049 for related primitive diagrams, A227849.

c := x^2*y+x^3*y+x*y+x*y^2+y^2+x^3+x^4: mc := p->expand((p*c-subs(x=0,p*c)-subs(y=0,p*c))/x/y): g2 := proc(n) option remember; global x,y,c,mc; expand((mc(g2(n-1))-subs(x=0,mc(g2(n-1))))/x-subs(x=0,g2(n-1))) end: g2(0) := 1: a := seq(subs(x=0,y=0,g2(n)),n=0..50);
A059710:=rsolve({(n+5)*(n+6)*A(n)=2*(n-1)*(2*n+5)*A(n-1)+(n-1)*(19*n+18)*A(n-2)+14*(n-1)*(n-2)*A(n-3),A(0)=1,A(1)=0,A(2)=1},A(n),makeproc);
# See Mihailovs reference for proof that this program is correct.
# Alec Mihailovs, Jun 17 2003

a[0] = 1; a[1] = 0; a[2] = 1; a[n_] := a[n] = (2*(n-1)*(2*n + 5)*a[n-1] + (n-1)*(19*n + 18)*a[n-2] + 14*(n-1)*(n-2)*a[n-3])/((n + 5)*(n + 6));
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Nov 17 2017 *)

{a(n) = if( n<1, n==0, (2*(n-1)*(2*n+5) * a(n-1) + (n-1)*(19*n+18) * a(n-2) + 14*(n-1)*(n-2) * a(n-3)) / ((n+5)*(n+6)))}; /* Michael Somos, Oct 28 2013 */

Limit_{n->oo} a(n+1)/a(n) = 7.
a(0)=1, a(1)=0, a(2)=1 and (n+5)*(n+6)*a(n) = 2*(n-1)*(2*n+5)*a(n-1)+(n-1)*(19*n+18)*a(n-2)+14*(n-1)*(n-2)*a(n-3) for n > 2. - Alec Mihailovs (alec(AT)mihailovs.com), Feb 12 2005
Let f(n) = a(n+3)*a(n+4)*a(n+5) - 15 * a(n+4)^2*a(n+3) ... - 2744 * a(n+2)*a(n+1)*a(n), a homogeneous cubic polynomial in {a(n), a(n+1), ..., a(n+5)} with 40 terms. Then f(n) = 0 unless n = -3. - Michael Somos, Nov 01 2013
Let g(n) = 30 * a(n+3)^2*a(n+4) - 450 * a(n+3)^4 ... - 76832 * a(n+2)*a(n+1)*a(n)^2, a homogeneous quartic polynomial in {a(n), a(n+1), ..., a(n+4)} with 56 terms. Then g(n) = 0 unless n = -3. - Michael Somos, Nov 01 2013
O.g.f.: -(1-7*x)^(4/3)*(x+1)^2*(1+2*x)^(2/3)*hypergeom([-2/3, 7/3],[2],-27*x*(x+1)^2/((1+2*x)*(7*x-1)^2))/(6*x^5)+(28*x^4+66*x^3+46*x^2+15*x+1)/(6*x^5). - Mark van Hoeij, Jul 26 2021

Removed "word" keyword because it is not appropriate. - Kang Seonghoon (lifthrasiir(AT)gmail.com), Oct 10 2008

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User: Greg Kuperberg

A060050 Number of irreducible nonpositively curved triangulations of an n-gon: All internal vertices have at valence at least 6 and no diagonals of the n-gon are allowed.

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A060049 Triangulations of an n-gon such that each internal vertex has valence at least 6, i.e., nonpositively curved triangulations.

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A059710 Dimension of space of invariants of n-th tensor power of 7-dimensional irreducible representation of G_2. Also the number of n-leaf, otherwise trivalent graphs in a disk such that all faces have at least 6 sides.

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