A122722 Number of triangulations of Delta^2 x Delta^(k-1).

1, 6, 108, 4488, 376200, 58652640, 16119956160, 7519632382080, 5788821019685760, 7197150396467808000, 14206044114169232371200, 43903287397136367836697600, 210012592354755890839147008000, 1540026232221309103088828327116800, 17170286302440610680613970557956096000, 289015112280462271460535463614055526400000

Offset: 1

Author

Jonathan Vos Post, Oct 22 2006

Keywords

• nonn

Comments

The number of triangulations of Delta^2 x Delta^(k) is between alpha^(k^2) and beta*(k^2) where alpha = (27/16)^(1/4) ~ 1.13975 and beta = 6^(1/6) ~ 1.34800 [p. 10 of Santos's handwritten notes about "The Cayley trick"].

There are arithmetic errors in Santos's lecture notes "The Cayley trick". The same table gives lozenge tilings of k*Delta^2.

From Petros Hadjicostas, Sep 13 2019: (Start)

The first column (indexed by k) of the table on p. 9 in Santos' handwritten notes "The Cayley trick" is actually the sequence (A273464(k, k*(k-1)/2 + 1): k >= 1).

In later published papers, Santos (2004, 2005) mentions that the number of triangulations of Delta^2 x Delta^k grows as exp(A244996*k^2/2 + o(k^2)) as k -> infinity. Notice that exp(A244996 * k^2/2) = A242710^(k^2/2). [See Theorem 1 and Theorem 4.9. Probably Theorem 1, part (2), in Santos (2004) has a typo.]

Note that alpha = (27/16)^(1/4) ~ 1.13975 < A242710^(k^2/2) ~ 1.175311 < beta = 6^(1/6) ~ 1.34800 (where alpha and beta are given on the first paragraph of these comments).

The reason the name of the sequence has "Delta^2 x Delta^(k-1)" rather than "Delta^2 x Delta^k" is because (according to Santos) the number of triangulations of Delta^2 x Delta^(k-1) equals k! times the number of lozenge tilings of k*Delta^2. (End)

Examples

			a(1) = 1 * 1! = 1.
a(2) = 3 * 2! = 6.
a(3) = 18 * 3! = 108.
a(4) = "187 * 4! = 2244" [sic]; actually 187 * 4! = 4488.
a(5) = "3135 * 5! = 188100" [sic]; actually 3135 * 5! = 376200.
		

Links

• J. A de Loera, Nonregular triangulations of products of simplices, Discrete Comp. Geom., 15(3) (1996), 253-264. [It may be related to this sequence.]
• J. A. de Loera, J. Rambau, and Francisco Santos, MSRI Summer School on Triangulations of point sets, Applications, Structures and Algorithms.
• J. A. De Loera, J. Rambau, and Francisco Santos, Further topics, in: Triangulations, vol 25 of Algor. Computat. Math. (2010), pp. 433-511.
• R. J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.
• Francisco Santos, The Cayley trick, handwritten lecture notes; see table on p. 9.
• Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004; see Theorem 1 (p. 2).
• Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), pp. 151-177.
• Benjamin Frederik Schröter, Matroidal subdivisions, Dressians and tropical Grassmannians, Ph.D. Dissertation, Technische Universität Berlin, Berlin, 2018; see Appendix on p. 111.

Crossrefs

Cf. A000124, A011555, A011556, A045943, A242710, A244996, A273464, A326367, A326368, A326369.

Formula

Conjectures: a(n) = n! * A273464(n, n*(n+1)/2) for n >= 1; a(n) = A011555(n-1) for n >= 2. [A273464(n,k) is defined for n >= 1 and 0 <= k <= n*(n+1)/2.] - Petros Hadjicostas, Sep 12 2019

Extensions

More terms (using the references) from Petros Hadjicostas, Sep 12 2019