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

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

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

See also Polygonal numbers#Order of basis

For a subset of nonnegative integers $S$, its order as a basis is defined as the minimal number $g$ such that all integers are representable as at least $g$ members of $S$.

Whereas the $k$-gonal numbers form a basis of order $k$ (the polygonal number theorem), and the $d$-dimensional hypercubic numbers form a basis of finite order dependent on $d$ (the Hilbert-Waring theorem), relatively little is known about the simplicial numbers. Pollock's Conjecture states that the tetrahedral numbers are a basis of order 5. A000797 gives the integers known not to be writeable as a sum of four, and A282172(n) is the number of ways to write $n$ as an ordered sum of five.

5 is the first number not a sum of 2 triangular numbers, 17 the first not a sum of 4 tetrahedrals, 64 the first not a sum of 7 pentachorals, 220 the first not a sum of 9 hexaterals, 839 the first not a sum of 12 heptapetals. These first six terms suggest the smallest number not writeable as $\mathrm {ord} (P_{d+1}^{(d)})-1$ $d$-simplexes is $A366978(d)=\sum _{j=1}^{d}P_{d+2}^{(d+1)}\left(\left\lfloor {\frac {d}{j}}\right\rfloor \right)=\sum _{j=1}^{d}{d+\left\lfloor {\frac {d}{j}}\right\rfloor \choose d+1}$, but this pattern breaks down with 3137 being the first not writeable as 14 octahexal numbers.

Differences

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

Partial sums

$\sum _{n=1}^{m}P_{d+1}^{(d)}(n)=P_{d+2}^{(d+1)}(n)\,$

Partial sums of reciprocals

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

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 $\sum _{n=1}^{m}{1 \over {P_{d+1}^{(d)}(n)}}=$ $d-{\frac {m+d}{m+d \choose d}} \over d-1$	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.