A011555 -id:A011555 - cp's OEIS frontend

A273464 The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n(n + 1)/2, read by rows.

1, 1, 3, 1, 9, 24, 18, 1, 18, 126, 434, 762, 630, 187, 1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135, 1, 45, 915, 11127, 90270, 515970, 2139120, 6523428, 14683401, 24256853, 28975770, 24383838, 13860321, 4966929, 989970, 81462, 1, 63

Offset: 1

R. J. Mathar, May 23 2016

			Triangle T(n,k) (with rows n >= 1 and columns k >= 0) begins as follows:
  1;
  1,  3;
  1,  9,  24,   18;
  1, 18, 126,  434,   762,   630,   187;
  1, 30, 387, 2814, 12699, 36894, 69242, 81936, 57672, 21432, 3135;
  ...

R. J. Mathar, Table of n, a(n) for n = 1..575
J. A. De Loera, J. Rambau, F. Santos, Further topics, in: Triangulations, vol 25 of Algor. Computat. Math. (2010), 433-511.
R. J. Mathar, Lozenge tilings of the equilateral triangle, arXiv:1909.06336 [math.CO], 2019.
Francisco Santos, The Cayley trick and triangulations of products of simplices, arXiv:math/0312069 [math.CO], 2004.
Francisco Santos, The Cayley trick and triangulations of products of simplices, Cont. Math. 374 (2005), 151-177.
Wikipedia, Lozenge.

Cf. A045943 (column k=1), A011555, A011556, A011781, A122722, A326367 (k=2), A326368 (k=3), A326369 (k=4), A000124 (row lengths).

T(n,2) = 3*(n-1)*(n-2)*(3*n^2+3*n-4)/8 . - R. J. Mathar, May 24 2016
T(n,3) = (n-2)*(9*n^5-9*n^4-81*n^3+81*n^2+160*n-192)/16. - Greg Dresden, Jul 03 2019
Conjecture: T(n,4) = 3*(n-2)*(n-3)*(9*n^6+9*n^5-135*n^4-81*n^3+670*n^2+104*n-1216)/128. - Greg Dresden, Jul 03 2019
Conjecture: T(n,5) = 3*(n-3)*(n+3)* (27*n^8 -135*n^7 -387*n^6 +2835*n^5 -168*n^4 -18732*n^3 +19568*n^2 +36992*n -56320)/1280. - R. J. Mathar, Jul 07 2019
From Petros Hadjicostas, Sep 13 2019: (Start)
Conjecture for rightmost terms: A122722(n) =  n! * T(n, n*(n+1)/2) for n >= 1.
Conjectures for column k >= 0: Sum_{0 <= s <= 2*k + 1} (-1)^s * binomial(2*k+1, s) * T(n-s, k) = 0 for n >= 2*k+2.
Sum_{0 <= s <= 2*k} (-1)^s * binomial(2*k, s) * T(n-s, k) = A011781(k) for n >= 2*k+1. (End)

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

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Oct 22 2006

nonn

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)

			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.

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.

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

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

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

cp's OEIS Frontend

A273464 The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n(n + 1)/2, read by rows.

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

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A011556 Number of regular triangulations of triangle X n-simplex.

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cp's OEIS Frontend

A273464 The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2*k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n*(n + 1)/2, read by rows.

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

Views

Author

Keywords

Comments

Examples

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Crossrefs

Formula

Extensions

A011556 Number of regular triangulations of triangle X n-simplex.

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A273464 The number of tilings of an equilateral triangle of side length n with k lozenges and n^2 - 2k unit triangles. Triangle T(n, k) with n >= 1 and 0 <= k <= n(n + 1)/2, read by rows.