Simplicial polytopic numbers

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

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 n^th simplicial d-polytopic numbers are given by the formulae :

$P_{d+1}^{(d)}(n)={\binom {d+(n-1)}{d}}={\frac {d^{(n)}}{d!}}$, or

$P_{d+1}^{(d)}(n)={\binom {n+(d-1)}{d}}={\frac {n^{(d)}}{d!}}=\left(\!\!{\binom {n}{d}}\!\!\right),$

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

$P_{d+1}^{(d)}(n)=P_{d+1}^{(d)}(n-1)+P_{d}^{(d-1)}(n)\,$

Generating function

$G_{\{P_{d+1}^{(d)}\}}(x)={\frac {x}{(1-x)^{d+1}}}$

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 d^th column of Pascal's triangle gives the d-dimensional simplicial polytopic numbers, forming simplex polytopes, 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. 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 $\scriptstyle k\geq 3$, 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 $\scriptstyle d\geq 2\,$. 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

$P_{d+1}^{(d)}(n)-P_{d+1}^{(d)}(n-1)=P_{d}^{(d-1)}(n)\,$

Partial sums

$\sum _{n=1}^{m}P_{d+1}^{(d)}(n)=?\,$

Partial sums of reciprocals

$\sum _{n=1}^{m}{\frac {1}{P_{d+1}^{(d)}(n)}}=?\,$

Sum of reciprocals

The sum of reciprocals ${\sum _{n=1}^{\infty }{1 \over {P_{d+1}^{(d)}(n)}}}={\frac {d}{d-1}}={\frac {1}{1-{\frac {1}{d}}}}$ 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:

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

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"

$N_{j}={\binom {d+1}{j+1}},\ (0\leq j\leq d)\,$

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	Formulae $P_{d+1}^{(d)}(n)\,$	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)
1	Triangular gnomon 1-simplex di-0-cell (2) {}	${\binom {n}{1}}$ ${\frac {n^{(1)}}{1!}}$	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}	${\binom {n+1}{2}}$ ${\frac {n^{(2)}}{2!}}$	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}	${\binom {n+2}{3}}$ ${\frac {n^{(3)}}{3!}}$	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}	${\binom {n+3}{4}}$ ${\frac {n^{(4)}}{4!}}$	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}	${\binom {n+4}{5}}$ ${\frac {n^{(5)}}{5!}}$	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}	${\binom {n+5}{6}}$ ${\frac {n^{(6)}}{6!}}$	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}	${\binom {n+6}{7}}$ ${\frac {n^{(7)}}{7!}}$	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}	${\binom {n+7}{8}}$ ${\frac {n^{(8)}}{8!}}$	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}	${\binom {n+8}{9}}$ ${\frac {n^{(9)}}{9!}}$	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}	${\binom {n+9}{10}}$ ${\frac {n^{(10)}}{10!}}$	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}	${\binom {n+10}{11}}$ ${\frac {n^{(11)}}{11!}}$	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}	${\binom {n+11}{12}}$ ${\frac {n^{(12)}}{12!}}$	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	Generating function $G_{\{P_{d+1}^{(d)}\}}(x)=\,$ ${\frac {x}{(1-x)^{d+1}}}\,$	Order of basis $g_{\{P_{d+1}^{(d)}\}}=\,$ $N_{0}+?\,$	Differences $P_{d+1}^{(d)}(n)-\,$ $P_{d+1}^{(d)}(n-1)=\,$ $P_{d}^{(d-1)}(n)\,$	Partial sums $\sum _{n=1}^{m}{P_{d+1}^{(d)}(n)}=$ $P_{d+2}^{(d+1)}(m)\,$	Partial sums of reciprocals $\sum _{n=1}^{m}{1 \over {P_{d+1}^{(d)}(n)}}=$ $d{\binom {m+d}{d}}-m-d \over (d-1){\binom {m+d}{d}}$	Sum of reciprocals $\sum _{n=1}^{\infty }{1 \over {P_{d+1}^{(d)}(n)}}=\,$ ${\frac {d}{d-1}}\,$
1	Triangular gnomon 1-simplex bi-0-cell (2) {}	${\frac {x}{(1-x)^{2}}}\,$	$1\,$ $(2d-1)\,$	$\,$	$\,$	$H_{m}\,$ [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] = $2m \over (m+1)$	$2\,$
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] = $3(m^{2}+3m) \over 2(m^{2}+3m+2)$	${\frac {3}{2}}$ [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] = $4(m^{3}+6m^{2}+11m) \over 3(m^{3}+6m^{2}+11m+6)$ 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	$P_{d+1}^{(d)}(n),\ n\geq 0\,$ 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

External links

• S. Plouffe, Approximations de Séries Génératrices et Quelques Conjectures, Dissertation, Université du Québec à Montréal, 1992.
• S. Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.
• Herbert S. Wilf, generatingfunctionology, 2^nd ed., 1994.