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Simplicial polytopic numbers

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The simplicial polytopic numbers are a family of sequences of figurate numbers corresponding to the d-dimensional simplex for each dimension d, where d is a nonnegative integer.


All figurate numbers are accessible via this structured menu: Classifications of figurate numbers

Contents

Minimal nondegenerate polytopes in a d-dimensional Euclidean space, d ≥ 0

In a d-dimensional Euclidean space \scriptstyle \mathbb{R}^d\,, d ≥ 0, the minimal number of vertices d + 1 gives the simplest d-polytope (the d-simplex,) i.e.:

  • d = 0: the 0-simplex (having 1 vertex) is the point (the 1 (-1)-cell, with 1 null polytope as facet)
  • d = 1: the 1-simplex (having 2 vertices) is the triangular gnomon (the 2 0-cell, with 2 points as facets)
  • d = 2: the 2-simplex (having 3 vertices) is the trigon (triangle) (the 3 1-cell, with 3 segments as facets)
  • d = 3: the 3-simplex (having 4 vertices) is the tetrahedron (the 4 2-cell, with 4 faces as facets)
  • d = 4: the 4-simplex (having 5 vertices) is the pentachoron (the 5 3-cell, with 5 rooms as facets)
  • d = 5: the 5-simplex (having 6 vertices) is the hexateron (the 6 4-cell, with 6 4-cells as facets)
  • d = 6: the 6-simplex (having 7 vertices) is the heptapeton (the 7 5-cell, with 7 5-cells as facets)
  • d = 7: the 7-simplex (having 8 vertices) is the octahexon (the 8 6-cell, with 8 6-cells as facets)
  • d = 8: the 8-simplex (having 9 vertices) is the enneahepton (the 9 7-cell, with 9 7-cells as facets)
  • ...
  • d = d: the d-simplex (having d+1 vertices) is the d+1 (d-1)-cell, with d+1 (d-1)-cells as facets

Formulae

The nth simplicial d-polytopic numbers are given by the formulae [1] [2][3]:

P^{(d)}_{d+1}(n) = \binom{d+(n-1)}{d} = \frac{d^{(n)}}{d!}, or
P^{(d)}_{d+1}(n) = \binom{n+(d-1)}{d} = \frac{n^{(d)}}{d!} = \left(\!\!{\binom{n}{d}}\!\!\right),

where d ≥ 0 is the dimension and n-1 ≥ 0 is the number of nondegenerate layered simplices (n-1 = 0 giving a single dot, a degenerate simplex) of the d-dimensional regular convex simplicial polytope number (d-simplex number.)

Recurrence relation

P^{(d)}_{d+1}(n) = P^{(d)}_{d+1}(n-1) + P^{(d-1)}_{d}(n)\,

Generating function

G_{\{P^{(d)}_{d+1}\}}(x) = \frac{x}{(1-x)^{d+1}}

Simplicial polytopic numbers and Pascal's (rectangular) triangle columns

Pascal's triangle
n = 0 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1
7 1 7 21 35 35 21 7 1
8 1 8 28 56 70 56 28 8 1
9 1 9 36 84 126 126 84 36 9 1
10 1 10 45 120 210 252 210 120 45 10 1
11 1 11 55 165 330 462 462 330 165 55 11 1
12 1 12 66 220 495 792 924 792 495 220 66 12 1
d = 0 1 2 3 4 5 6 7 8 9 10 11 12


The dth column of Pascal's triangle gives the d-dimensional simplicial polytopic numbers, forming simplex polytopes,[4] where (-1)-cells correspond to the empty set, 0-cells are vertices, 1-cells are edges, 2-cells are faces, and so on...
 
d = 0 0-simplicial numbers Point numbers Form point (1 (-1)-cells "faces") (0-simplex)
d = 1 1-simplicial numbers Linear numbers Form segments (2 0-cells "faces") (1-simplex)
d = 2 2-simplicial numbers Triangular numbers Form triangles (3 1-cells "faces") (2-simplex)
d = 3 3-simplicial numbers Tetrahedral numbers Form tetrahedrons (4 2-cells "faces") (3-simplex)
d = 4 4-simplicial numbers Pentachoron numbers Form pentachorons (5 3-cells "faces") (4-simplex)
d = 5 5-simplicial numbers Hexateron numbers Form hexaterons (6 4-cells "faces") (5-simplex)
d = 6 6-simplicial numbers Heptapeton numbers Form heptapetons (7 5-cells "faces") (6-simplex)
d = 7 7-simplicial numbers Octahexon numbers Form octahexons (8 6-cells "faces") (7-simplex)
d = 8 8-simplicial numbers Nonahepton numbers Form nonaheptons (9 7-cells "faces") (8-simplex)

Order of basis

In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and k k-polygonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found.[5] Joseph Louis Lagrange proved the square case (known as the four squares theorem) in 1770 and Gauss proved the triangular case in 1796. In 1813, Cauchy finally proved the horizontal generalization that every nonnegative integer can be written as a sum of k k-gon numbers (known as the polygonal number theorem,) while a vertical (higher dimensional) generalization has also been made (known as the Hilbert-Waring problem.)

A nonempty subset \scriptstyle A\, of nonnegative integers is called a basis of order \scriptstyle g\, if \scriptstyle g\, is the minimum number with the property that every nonnegative integer can be written as a sum of \scriptstyle g\, elements in \scriptstyle A\,. Lagrange’s sum of four squares can be restated as the set \scriptstyle \{n^2|n = 0, 1, 2, \ldots\}\, of nonnegative squares forms a basis of order 4.

Theorem (Cauchy) For every \scriptstyle k \ge 3, the set \scriptstyle \{P(k, n)|n = 0, 1, 2, \ldots\}\, of k-gon numbers forms a basis of order \scriptstyle k\,, i.e. every nonnegative integer can be written as a sum of \scriptstyle k\, k-gon numbers.

We note that polygonal numbers are two dimensional analogues of squares. Obviously, cubes, fourth powers, fifth powers, ... are higher dimensional analogues of squares. In 1770, Waring stated without proof that every nonnegative integer can be written as a sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 1909, Hilbert proved that there is a finite number \scriptstyle g(d)\, such that every nonnegative integer is a sum of \scriptstyle g(d)\, \scriptstyle d\,th powers, i.e. the set \scriptstyle \{n^d|n = 0, 1, 2, \ldots\}\, of \scriptstyle d\,th powers forms a basis of order \scriptstyle g(d)\,. The Hilbert-Waring problem is concerned with the study of \scriptstyle g(d)\, for \scriptstyle d \ge 2\,. This problem was one of the most important research topics in additive number theory in last 90 years, and it is still a very active area of research.

Differences

P^{(d)}_{d+1}(n) - P^{(d)}_{d+1}(n-1) = P^{(d-1)}_{d}(n)\,

Partial sums

\sum_{n=1}^m P^{(d)}_{d+1}(n) = ?\,

Partial sums of reciprocals

\sum_{n=1}^m \frac{1}{P^{(d)}_{d+1}(n)} = ?\,

Sum of reciprocals

The sum of reciprocals {\sum_{n=1}^\infty {1\over{P^{(d)}_{d+1}(n)}}} = \frac{d}{d-1} = \frac{1}{1-\frac{1}{d}} can be interpreted as \frac{1}{p}, where p = {1-\frac{1}{d}} is the probability that d does not divide a random integer x or the probability that two random integers x and y have different residues modulo d.

The reciprocal of the sum of reciprocals thus gives:

\frac{1}{\sum_{n=1}^\infty {1\over{P^{(d)}_{d+1}(n)}}} = \frac{d-1}{d} = {1-\frac{1}{d}} = {P(x \not\equiv 0 \pmod{d})} = {P(x\not\equiv y \pmod {d})}

Graphically, the second interpretation is the probability that a 2-D integer lattice point (x,y)\, does not lay on any straight line of the family y=x+kd\,, for integer values of k. This family of functions consists of the bisection of the first and third quadrants and all its parallels at distances \frac{kd}{\sqrt{2}}. When d=1\, this includes all lattice points, hence p=0\,.

Number of j-dimensional "vertices"

N_j = \binom{d+1}{j+1},\ (0 \le j \le d)\,

Table of formulae and values

N0, N1, N2,N3, ... are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional), cells (3-dimensional)... respectively, where the (n-1)-dimensional "vertices" are the actual facets. The simplicial polytopic numbers are listed by increasing number N0 of vertices.

Simplicial numbers formulae and values
d Name

d-simplex

d+1 (d-1)-cell

(N0, N1, N2, ...)

Schläfli symbol[6]

Formulae

P^{(d)}_{d+1}(n)\,

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 OEIS

number

0 Point

0-simplex

hena-(-1)-cell

()

{}

\binom{n-1}{0} 0 1 1 1 1 1 1 1 1 1 1 1 1 (NOT A057427) [7]
1 Triangular gnomon

1-simplex

di-0-cell

(2)

{}

\binom{n}{1}

\frac{n^{(1)}}{1!}[2]

0 1 2 3 4 5 6 7 8 9 10 11 12 A000027(n)
2 Triangular

2-simplex

tri-1-cell

(3, 3)

{3}

\binom{n+1}{2}

\frac{n^{(2)}}{2!}[2]

0 1 3 6 10 15 21 28 36 45 55 66 78 A000217(n)
3 Tetrahedral

3-simplex

tetra-2-cell

(4, 6, 4)

{3, 3}

\binom{n+2}{3}

\frac{n^{(3)}}{3!}[2]

0 1 4 10 20 35 56 84 120 165 220 286 364 A000292(n)
4 Pentachoron

4-simplex

penta-3-cell

(5, 10, 10, 5)

{3, 3, 3}

\binom{n+3}{4}

\frac{n^{(4)}}{4!}[2]

0 1 5 15 35 70 126 210 330 495 715 1001 1365 A000332(n+3)
5 Hexateron

5-simplex

hexa-4-cell

(6, 15, 20, 15, 6)

{3, 3, 3, 3}

\binom{n+4}{5}

\frac{n^{(5)}}{5!}[2]

0 1 6 21 56 126 252 462 792 1287 2002 3003 4368 A000389(n+4)
6 Heptapeton

6-simplex

hepta-5-cell

(7, 21, 35, 35, 21, 7)

{3, 3, 3, 3, 3}

\binom{n+5}{6}

\frac{n^{(6)}}{6!}[2]

0 1 7 28 84 210 462 924 1716 3003 5005 8008 12376 A000579(n+5)
7 Octahexon

7-simplex

octa-6-cell

(8, 28, 56, 70, 56, 28, 8)

{3, 3, 3, 3, 3, 3}

\binom{n+6}{7}

\frac{n^{(7)}}{7!}[2]

0 1 8 36 120 330 792 1716 3432 6435 11440 19448 31824 A000580(n+6)
8 Enneahepton

8-simplex

nona-7-cell

(9, 36, 84, 126, 126, 84, 36, 9)

{3, 3, 3, 3, 3, 3, 3}

\binom{n+7}{8}

\frac{n^{(8)}}{8!}[2]

0 1 9 45 165 495 1287 3003 6435 12870 24310 43758 75582 A000581(n+7)
9 Decaocton

9-simplex

deca-8-cell

(10, 45, 120, 210, 252, 210, 120, 45, 10)

{3, 3, 3, 3, 3, 3, 3, 3}

\binom{n+8}{9}

\frac{n^{(9)}}{9!}[2]

0 1 10 55 220 715 2002 5005 11440 24310 48620 92378 167960 A000582(n+8)
10 Hendecaenneon

10-simplex

hendeca-9-cell

(11, 55, 165, 330, 462, 462, 330, 165, 55, 11)

{3, 3, 3, 3, 3, 3, 3, 3, 3}

\binom{n+9}{10}

\frac{n^{(10)}}{10!}[2]

0 1 11 66 286 1001 3003 8008 19448 43758 92378 184756 352716 A001287(n+9)
11 Dodecadecon

11-simplex

dodeca-10-cell

(12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12)

{3, 3, 3, 3, 3, 3, 3, 3, 3, 3}

\binom{n+10}{11}

\frac{n^{(11)}}{11!}[2]

0 1 12 78 364 1365 4368 12376 31824 75582 167960 352716 705432 A001288(n+10)
12 Tridecahendecon

12-simplex

trideca-11-cell

(13, ... Pascal's triangle 13th row..., 13)

{3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}

\binom{n+11}{12}

\frac{n^{(12)}}{12!}[2]

0 1 13 91 455 1820 6188 18564 50388 125970 293930 646646 1352078 A010965(n+11)


Table of related formulae and values

N0, N1, N2,N3, ... are the number of vertices (0-dimensional), edges (1-dimensional), faces (2-dimensional), cells (3-dimensional)... respectively, where the (n-1)-dimensional "vertices" are the actual facets. The simplicial polytopic numbers are listed by increasing number N0 of vertices.

Simplicial numbers related formulae and values
d Name

d-simplex

d+1 (d-1)-cell

(N0, N1, N2, ...)

Schläfli symbol[6]

Generating

function

G_{\{P^{(d)}_{d+1}\}}(x) =\,


\frac{x}{(1-x)^{d+1}}\,

Order

of basis

g_{\{P^{(d)}_{d+1}\}} =\,


N_0 + ?\, [8][9][10]

Differences

P^{(d)}_{d+1}(n) - \,

P^{(d)}_{d+1}(n-1) =\,


P^{(d-1)}_{d}(n)\,

Partial sums

\sum_{n=1}^m {P^{(d)}_{d+1}(n)} =


P^{(d+1)}_{d+2}(m)\,

Partial sums of reciprocals

\sum_{n=1}^m {1\over{P^{(d)}_{d+1}(n)}} =


d \binom{m+d}{d}-m-d \over (d-1) \binom{m+d}{d}

Sum of reciprocals[11]

\sum_{n=1}^\infty {1\over{P^{(d)}_{d+1}(n)}} =\,


\frac{d}{d-1}\,

1 Triangular gnomon

1-simplex

bi-0-cell

(2)

{}

\frac{x}{(1-x)^{2}}\, 1\,


(2d-1)\,

\, \, H_m\, [12] [1]


A001008(m)/A002805(m) (reduced)


A000254(m)/A000142(m)

\lim_{m \to \infty} H_m \sim log(m) \to \infty\,
2 Triangular

2-simplex

tri-1-cell

(3, 3)

{3}

\frac{x}{(1-x)^{3}}\, 3\,


(2d-1)\,

\, \, 2 \binom{m+2}{2}-m-2 \over 1 \binom{m+2}{2} [2] =


2 m \over (m+1)

2\,
3 Tetrahedral

3-simplex

tetra-2-cell

(4, 6, 4)

{3, 3}

\frac{x}{(1-x)^{4}}\, 5?\,


(2d-1)?\,

\, \, 3 \binom{m+3}{3}-m-3 \over 2 \binom{m+3}{3} [3] =


3(m^2+3 m) \over 2(m^2+3 m+2)

\frac{3}{2} [4]
4 Pentachoron

4-simplex

penta-3-cell

(5, 10, 10, 5)

{3, 3, 3}

\frac{x}{(1-x)^{5}}\, 8?\,


(2d)?\,

\, \, 4 \binom{m+4}{4}-m-4 \over 3 \binom{m+4}{4} [5] =


4 (m^3+6 m^2+11 m) \over 3 (m^3+6 m^2+11 m+6)


A118411/A118412

\frac{4}{3}
5 Hexateron

5-simplex

hexa-4-cell

(6, 15, 20, 15, 6)

{3, 3, 3, 3}

\frac{x}{(1-x)^{6}}\, 10?\,


(2d)?\,

\, \, 5 \binom{m+5}{5}-m-5 \over 4 \binom{m+5}{5} [6] \frac{5}{4}
6 Heptapeton

6-simplex

hepta-5-cell

(7, 21, 35, 35, 21, 7)

{3, 3, 3, 3, 3}

\frac{x}{(1-x)^{7}}\, 13?\,


(2d+1)?\,

\, \, 6 \binom{m+6}{6}-m-6 \over 5 \binom{m+6}{6} [7] \frac{6}{5}
7 Octahexon

7-simplex

octa-6-cell

(8, 28, 56, 70, 56, 28, 8)

{3, 3, 3, 3, 3, 3}

\frac{x}{(1-x)^{8}}\, 15?\,


(2d+1)?\,

\, \, 7 \binom{m+7}{7}-m-7 \over 6 \binom{m+7}{7} [8] \frac{7}{6}
8 Enneahepton

8-simplex

nona-7-cell

(9, 36, 84, 126, 126, 84, 36, 9)

{3, 3, 3, 3, 3, 3, 3}

\frac{x}{(1-x)^{9}}\, (2d+1)?\, \, \, 8 \binom{m+8}{8}-m-8 \over 7 \binom{m+8}{8} [9] \frac{8}{7}
9 Decaocton

9-simplex

deca-8-cell

(10, 45, 120, 210, 252, 210, 120, 45, 10)

{3, 3, 3, 3, 3, 3, 3, 3}

\frac{x}{(1-x)^{10}}\, (2d+1)?\, \, \, 9 \binom{m+9}{9}-m-9 \over 8 \binom{m+9}{9} [10] \frac{9}{8}
10 Hendecaenneon

10-simplex

hendeca-9-cell

(11, 55, 165, 330, 462, 462, 330, 165, 55, 11)

{3, 3, 3, 3, 3, 3, 3, 3, 3}

\frac{x}{(1-x)^{11}}\, (2d+1)?\, \, \, 10 \binom{m+10}{10}-m-10 \over 9 \binom{m+10}{10} [11] \frac{10}{9}
11 Dodecadecon

11-simplex

dodeca-10-cell

(12, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12)

{3, 3, 3, 3, 3, 3, 3, 3, 3, 3}

\frac{x}{(1-x)^{12}}\, (2d+1)?\, \, \, 11 \binom{m+11}{11}-m-11 \over 10 \binom{m+11}{11} [12] \frac{11}{10}
12 Tridecahendecon

12-simplex

trideca-11-cell

(13, ... Pascal's triangle 13th row..., 13)

{3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}

\frac{x}{(1-x)^{13}}\, (2d+1)?\, \, \, 12 \binom{m+12}{12}-m-12 \over 11 \binom{m+12}{12} [13] \frac{12}{11}


Table of sequences

Simplicial polytopic numbers sequences
d P^{(d)}_{d+1}(n),\ n \ge 0\, sequences
1 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, ...}
2 {0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, ...}
3 {0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, ...}
4 {0, 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, ...}
5 {0, 1, 6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 169911, ...}
6 {0, 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, ...}
7 {0, 1, 8, 36, 120, 330, 792, 1716, 3432, 6435, 11440, 19448, 31824, 50388, 77520, 116280, 170544, 245157, 346104, 480700, 657800, 888030, 1184040, 1560780, 2035800, ...}
8 {0, 1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310, 43758, 75582, 125970, 203490, 319770, 490314, 735471, 1081575, 1562275, 2220075, 3108105, 4292145, 5852925, ...}
9 {0, 1, 10, 55, 220, 715, 2002, 5005, 11440, 24310, 48620, 92378, 167960, 293930, 497420, 817190, 1307504, 2042975, 3124550, 4686825, 6906900, 10015005, 14307150, ...}
10 {0, 1, 11, 66, 286, 1001, 3003, 8008, 19448, 43758, 92378, 184756, 352716, 646646, 1144066, 1961256, 3268760, 5311735, 8436285, 13123110, 20030010, 30045015, ... }
11 {0, 1, 12, 78, 364, 1365, 4368, 12376, 31824, 75582, 167960, 352716, 705432, 1352078, 2496144, 4457400, 7726160, 13037895, 21474180, 34597290, 54627300, ...}
12 {0, 1, 13, 91, 455, 1820, 6188, 18564, 50388, 125970, 293930, 646646, 1352078, 2704156, 5200300, 9657700, 17383860, 30421755, 51895935, 86493225, 141120525, ...}


See also

Centered simplicial polytopic numbers

Notes

  1. Where \scriptstyle P^{(d)}_{N_0}(n)\, is the d-dimensional regular convex polytope number with N0 0-dimensional elements (vertices V.)
  2. 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 Weisstein, Eric W., Rising Factorial, From MathWorld--A Wolfram Web Resource.
  3. Weisstein, Eric W., Multichoose, From MathWorld--A Wolfram Web Resource.
  4. Weisstein, Eric W., Simplex, From MathWorld--A Wolfram Web Resource.
  5. Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
  6. 6.0 6.1 Weisstein, Eric W., Schläfli Symbol, From MathWorld--A Wolfram Web Resource.
  7. A057427 is the sign function (-1 for n < 0, 0 for n = 0, +1 for n > 0,) while what we get here is the characteristic function of positive integers (0 for n ≤ 0, +1 for n ≥ 1.)
  8. Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
  9. HYUN KWANG KIM, ON REGULAR POLYTOPE NUMBERS.
  10. Pollock, Frederick, On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders, Abstracts of the Papers Communicated to the Royal Society of London, 5 (1850) pp. 922-924.
  11. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  12. Sondow, Jonathan and Weisstein, Eric W., Harmonic Number, From MathWorld--A Wolfram Web Resource.

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