A019502 -id:A019502

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A019503

Simplexity of the n-cube: minimal cardinality of triangulation of n-cube using n-simplices whose vertices are vertices of the n-cube.

+10
8

1, 2, 5, 16, 67, 308, 1493

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

5522 <= a(8) <= 11944 [Haiman, Ziegler]. - Jonathan Vos Post, Jul 13 2005

REFERENCES

H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, C9.

Warren D. Smith, Lower bounds for triangulations of the N-cube, manuscript, 1994.

Gunter M. Ziegler, Lectures on Polytopes, Revised First Edn., Graduate Texts in Mathematics, Springer, 1994, p. 147.

LINKS

Table of n, a(n) for n=1..7.

A. Glazyrin, Lower bounds for the simplexity of the n-cube, Discrete Math. 312 (2012), no. 24, 3656--3662. MR2979495. - From N. J. A. Sloane, Nov 07 2012

R. B. Hughes and M. R. Anderson, Simplexity of the cube, Discrete Mathematics, 158 (1996) 99-150, esp. p. 100.

Mark Haiman, A simple and relatively efficient triangulation of the n-cube, Discrete Comput. Geometry 6 (1991), 287-289.

D. Orden, F. Santos, Asymptotically efficient triangulations of the d-cube, Discr. Comput. Geom. 30 (2003) 509, Table 1.

Warren D. Smith, A lower bound for the simplexity of the n-cube via hyperbolic volumes, Combinatorics of polytopes. European J. Combin. 21 (2000), no. 1, 131-137. MR1737333 (2001c:52004).

Chuanming Zong, What is known about unit cubes, Bull. Amer. Math. Soc., 42 (2005), 181-211.

CROSSREFS

Other sequences dealing with different ways to attack this problem. They give further references: A019502, A019504, A166932, A166932, A239912, A275518.

KEYWORD

nonn,hard,nice,more

AUTHOR

N. J. A. Sloane

STATUS

approved

A019504

Number of simplices in minimal corner-slicing triangulation of n-cube.

+10
6

1, 2, 5, 16, 67, 324, 1820

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

H. T. Croft, K. J. Falconer and R. K. Guy, Unsolved Problems in Geometry, C9.

LINKS

Table of n, a(n) for n=1..7.

R. B. Hughes and M. R. Anderson, Simplexity of the cube, Discr. Math., 158 (1996), 99-150.

CROSSREFS

Cf. A019502, A019503.

KEYWORD

nonn,hard,nice,more

AUTHOR

N. J. A. Sloane

STATUS

approved

A275518

Number of simplices in corner-cut triangulation of the n-cube.

+10
6

1, 2, 5, 16, 67, 364, 2445, 19296, 173015, 1728604, 19011049, 228124384, 2965598547, 41518338684, 622774990133, 9964399645504, 169394793547567, 3049106282938684, 57933019373868897, 1158660387473183616, 24331868136927943019, 535301099012395872028

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

This corrects the value of a(10) in A239911 published by Sallee in Discr. Math. 40. The correct value is for example given by Lee.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..200

Carl W. Lee, Triangulating the d-cube, Annals of the New York Academy of Sciences 440 (1985): 205-211.

John F. Sallee, A note on minimal triangulations of an n-cube, Discrete Appl. Math. 4 (1982), no. 3, 211-215. MR0675850 (84g:52019)

John F. Sallee, The middle-cut triangulations of the n-cube, SIAM J. Algebraic Discrete Methods 5 (1984), no. 3, 407-419. MR0752044 (86c:05054). See Table 2.

John F. Sallee, A triangulation of the n-cube, Discrete Math. 40 (1982), no. 1, 81-86. MR0676714 (84d:05065b)

FORMULA

a(n) = 1 + 2^(n-1) - n! + n!*Sum_{i=1..n} (2^(i-1)-1)/i!. - Andrew Howroyd, Sep 06 2023, after Maple program

MAPLE

p := proc(d, x)

add( x^i/i!, i=0..d) ;

end proc:

A275518 := proc(d)

d!*(p(d, 2)/2-p(d, 1))+2^(d-1)-d!/2+1 ;

end proc:

seq(A275518(d), d=1..18) ;

MATHEMATICA

p[d_, x_] := Sum[x^i/i!, {i, 0, d}];

A275518[d_] := d!*(p[d, 2]/2 - p[d, 1]) + 2^(d - 1) - d!/2 + 1;

Table[A275518[d], {d, 1, 18}] (* Jean-François Alcover, Sep 06 2023, after Maple program *)

PROG

(PARI) a(n) = 1 + 2^(n-1) - n! + n!*sum(i=1, n, (2^(i-1)-1)/i!) \\ Andrew Howroyd, Sep 06 2023

CROSSREFS

Cf. A239911, A019502, A019503, A019504, A166932.

KEYWORD

nonn,easy

AUTHOR

R. J. Mathar, Jul 31 2016

EXTENSIONS

Terms a(19) and beyond from Andrew Howroyd, Sep 06 2023

STATUS

approved

A166932

Lower bounds for minimal number of simplices in a triangulation of the n-dimensional cube (A019503).

+10
5

5, 16, 67, 308, 1493, 5522

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,1

COMMENTS

The terms are given in Table 1 on page 2 of the Glazyrin reference.

There are many lists of bounds in different papers which differ by range, values, and methods used to obtain them. - Andrey Zabolotskiy, Nov 17 2017

LINKS

Table of n, a(n) for n=3..8.

A. Bliss, F. E. Su, Lower bounds for simplicial covers and triangulations of cubes, arXiv:math/0310142 [math.CO], 2003 (see Table 1 page 4).

A. Bliss, F. E. Su. Lower bounds for simplicial covers and triangulations of cubes, Discrete Comput. Geom. 33 (2005), 669-686.

R. W. Cottle, Minimal triangulation of the 4-cube, Discrete Math., 40(1):25-29, 1982.

Alexey Glazyrin, Lower bounds for the simplexity of the n-cube, arXiv:0910.4200 [math.MG], 2009-2012 (see Table 1 page 2).

R. B. Hughes and M. R. Anderson, Simplexity of the cube, Discrete Math., 158(1-3):99-150, 1996.

CROSSREFS

Cf. A019502, A019503, A019504.

KEYWORD

nonn,less

AUTHOR

Jonathan Vos Post, Oct 24 2009

STATUS

approved

A239912

Number of simplices is middle-cut slicing of n-cube.

+10
2

1, 2, 5, 16, 67, 324, 1962, 13248, 106181, 931300

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..10.

John F. Sallee, The middle-cut triangulations of the n-cube. SIAM J. Algebraic Discrete Methods 5 (1984), no. 3, 407--419. MR0752044 (86c:05054).

CROSSREFS

Other sequences dealing with different ways to attack this problem. They give further references: A019502, A019503, A019504, A166932, A166932, A275518.

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane, Apr 09 2014

STATUS

approved

A239911

Erroneous version of A275518.

+10
1

1, 2, 5, 16, 67, 364, 2445, 19296, 173015, 1720924

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

Lee, Carl W. "TRIANGULATING THE d‐CUBE." Annals of the New York Academy of Sciences 440.1 (1985): 205-211.

Sallee, John F. A note on minimal triangulations of an n-cube. Discrete Appl. Math. 4 (1982), no. 3, 211--215. MR0675850 (84g:52019)

Sallee, John F. The middle-cut triangulations of the n-cube. SIAM J. Algebraic Discrete Methods 5 (1984), no. 3, 407--419. MR0752044 (86c:05054). See Table 2.

LINKS

Table of n, a(n) for n=1..10.

Sallee, John F. A triangulation of the n-cube, Discrete Math. 40 (1982), no. 1, 81--86. MR0676714 (84d:05065b)

CROSSREFS

Cf. A019502, A019503, A019504, A166932.

KEYWORD

dead

AUTHOR

N. J. A. Sloane, Apr 09 2014

STATUS

approved

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