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Generic Pascal's m-simplex

Let m (m > 0) be a number of terms of a polynomial and n (n ≥ 0) be a power the polynomial is raised to.

Let ⁠ ∧ {\displaystyle \wedge } ⁠m denote a Pascal's m-simplex. Each Pascal's m-simplex is a semi-infinite object, which consists of an infinite series of its components.

Let ⁠ ∧ {\displaystyle \wedge } ⁠mn denote its nth component, itself a finite (m − 1)-simplex with the edge length n, with a notational equivalent △ n m − 1 {\displaystyle \vartriangle _{n}^{m-1}} .

nth component

∧ n m = △ n m − 1 {\displaystyle \wedge _{n}^{m}=\vartriangle _{n}^{m-1}} consists of the coefficients of multinomial expansion of a polynomial with m terms raised to the power of n:

| x | n = ∑ | k | = n ( n k ) x k ;     x ∈ R m ,   k ∈ N 0 m ,   n ∈ N 0 ,   m ∈ N {\displaystyle |x|^{n}=\sum _{|k|=n}{{\binom {n}{k}}x^{k}};\ \ x\in \mathbb {R} ^{m},\ k\in \mathbb {N} _{0}^{m},\ n\in \mathbb {N} _{0},\ m\in \mathbb {N} }

where | x | = ∑ i = 1 m x i ,   | k | = ∑ i = 1 m k i ,   x k = ∏ i = 1 m x i k i . {\displaystyle \textstyle |x|=\sum _{i=1}^{m}{x_{i}},\ |k|=\sum _{i=1}^{m}{k_{i}},\ x^{k}=\prod _{i=1}^{m}{x_{i}^{k_{i}}}.}

Example for ⋀4

Pascal's 4-simplex (sequence A189225 in the OEIS), sliced along the k4. All points of the same color belong to the same nth component, from red (for n = 0) to blue (for n = 3).

Specific Pascal's simplices

Pascal's 1-simplex

⁠ ∧ {\displaystyle \wedge } ⁠1 is not known by any special name.

nth component

∧ n 1 = △ n 0 {\displaystyle \wedge _{n}^{1}=\vartriangle _{n}^{0}} (a point) is the coefficient of multinomial expansion of a polynomial with 1 term raised to the power of n:

( x 1 ) n = ∑ k 1 = n ( n k 1 ) x 1 k 1 ;     k 1 , n ∈ N 0 {\displaystyle (x_{1})^{n}=\sum _{k_{1}=n}{n \choose k_{1}}x_{1}^{k_{1}};\ \ k_{1},n\in \mathbb {N} _{0}}

Arrangement of △ n 0 {\displaystyle \vartriangle _{n}^{0}}

( n n ) {\displaystyle \textstyle {n \choose n}}

which equals 1 for all n.

Pascal's 2-simplex

∧ 2 {\displaystyle \wedge ^{2}} is known as Pascal's triangle (sequence A007318 in the OEIS).

nth component

∧ n 2 = △ n 1 {\displaystyle \wedge _{n}^{2}=\vartriangle _{n}^{1}} (a line) consists of the coefficients of binomial expansion of a polynomial with 2 terms raised to the power of n:

( x 1 + x 2 ) n = ∑ k 1 + k 2 = n ( n k 1 , k 2 ) x 1 k 1 x 2 k 2 ;     k 1 , k 2 , n ∈ N 0 {\displaystyle (x_{1}+x_{2})^{n}=\sum _{k_{1}+k_{2}=n}{n \choose k_{1},k_{2}}x_{1}^{k_{1}}x_{2}^{k_{2}};\ \ k_{1},k_{2},n\in \mathbb {N} _{0}}

Arrangement of △ n 1 {\displaystyle \vartriangle _{n}^{1}}

( n n , 0 ) , ( n n − 1 , 1 ) , … , ( n 1 , n − 1 ) , ( n 0 , n ) {\displaystyle \textstyle {n \choose n,0},{n \choose n-1,1},\ldots ,{n \choose 1,n-1},{n \choose 0,n}}

Pascal's 3-simplex

∧ 3 {\displaystyle \wedge ^{3}} is known as Pascal's tetrahedron (sequence A046816 in the OEIS).

nth component

∧ n 3 = △ n 2 {\displaystyle \wedge _{n}^{3}=\vartriangle _{n}^{2}} (a triangle) consists of the coefficients of trinomial expansion of a polynomial with 3 terms raised to the power of n:

( x 1 + x 2 + x 3 ) n = ∑ k 1 + k 2 + k 3 = n ( n k 1 , k 2 , k 3 ) x 1 k 1 x 2 k 2 x 3 k 3 ;     k 1 , k 2 , k 3 , n ∈ N 0 {\displaystyle (x_{1}+x_{2}+x_{3})^{n}=\sum _{k_{1}+k_{2}+k_{3}=n}{n \choose k_{1},k_{2},k_{3}}x_{1}^{k_{1}}x_{2}^{k_{2}}x_{3}^{k_{3}};\ \ k_{1},k_{2},k_{3},n\in \mathbb {N} _{0}}

Arrangement of △ n 2 {\displaystyle \vartriangle _{n}^{2}}

( n n , 0 , 0 ) , ( n n − 1 , 1 , 0 ) , … … , ( n 1 , n − 1 , 0 ) , ( n 0 , n , 0 ) ( n n − 1 , 0 , 1 ) , ( n n − 2 , 1 , 1 ) , … … , ( n 0 , n − 1 , 1 ) ⋮ ( n 1 , 0 , n − 1 ) , ( n 0 , 1 , n − 1 ) ( n 0 , 0 , n ) {\displaystyle {\begin{aligned}\textstyle {n \choose n,0,0}&,\textstyle {n \choose n-1,1,0},\ldots \ldots ,{n \choose 1,n-1,0},{n \choose 0,n,0}\\\textstyle {n \choose n-1,0,1}&,\textstyle {n \choose n-2,1,1},\ldots \ldots ,{n \choose 0,n-1,1}\\&\vdots \\\textstyle {n \choose 1,0,n-1}&,\textstyle {n \choose 0,1,n-1}\\\textstyle {n \choose 0,0,n}\end{aligned}}}

Properties

Inheritance of components

∧ n m = △ n m − 1 {\displaystyle \wedge _{n}^{m}=\vartriangle _{n}^{m-1}} is numerically equal to each (m − 1)-face (there is m + 1 of them) of △ n m = ∧ n m + 1 {\displaystyle \vartriangle _{n}^{m}=\wedge _{n}^{m+1}} , or:

∧ n m = △ n m − 1 ⊂   △ n m = ∧ n m + 1 {\displaystyle \wedge _{n}^{m}=\vartriangle _{n}^{m-1}\subset \ \vartriangle _{n}^{m}=\wedge _{n}^{m+1}}

From this follows, that the whole ∧ m {\displaystyle \wedge ^{m}} is (m + 1)-times included in ∧ m + 1 {\displaystyle \wedge ^{m+1}} , or:

∧ m ⊂ ∧ m + 1 {\displaystyle \wedge ^{m}\subset \wedge ^{m+1}}

Example

	∧ 1 {\displaystyle \wedge ^{1}}	∧ 2 {\displaystyle \wedge ^{2}}	∧ 3 {\displaystyle \wedge ^{3}}	∧ 4 {\displaystyle \wedge ^{4}}
∧ 0 m {\displaystyle \wedge _{0}^{m}}	1	1	1	1
∧ 1 m {\displaystyle \wedge _{1}^{m}}	1	1 1	1 1 1	1 1 1 1
∧ 2 m {\displaystyle \wedge _{2}^{m}}	1	1 2 1	1 2 1 2 2 1	1 2 1 2 2 1 2 2 2 1
∧ 3 m {\displaystyle \wedge _{3}^{m}}	1	1 3 3 1	1 3 3 1 3 6 3 3 3 1	1 3 3 1 3 6 3 3 3 1 3 6 3 6 6 3 3 3 3 1

For more terms in the above array refer to (sequence A191358 in the OEIS)

Equality of sub-faces

Conversely, ∧ n m + 1 = △ n m {\displaystyle \wedge _{n}^{m+1}=\vartriangle _{n}^{m}} is (m + 1)-times bounded by △ n m − 1 = ∧ n m {\displaystyle \vartriangle _{n}^{m-1}=\wedge _{n}^{m}} , or:

∧ n m + 1 = △ n m ⊃ △ n m − 1 = ∧ n m {\displaystyle \wedge _{n}^{m+1}=\vartriangle _{n}^{m}\supset \vartriangle _{n}^{m-1}=\wedge _{n}^{m}}

From this follows, that for given n, all i-faces are numerically equal in nth components of all Pascal's (m > i)-simplices, or:

∧ n i + 1 = △ n i ⊂ △ n m > i = ∧ n m > i + 1 {\displaystyle \wedge _{n}^{i+1}=\vartriangle _{n}^{i}\subset \vartriangle _{n}^{m>i}=\wedge _{n}^{m>i+1}}

Example

The 3rd component (2-simplex) of Pascal's 3-simplex is bounded by 3 equal 1-faces (lines). Each 1-face (line) is bounded by 2 equal 0-faces (vertices):

2-simplex 1-faces of 2-simplex 0-faces of 1-face 1 3 3 1 1 . . . . . . 1 1 3 3 1 1 . . . . . . 1 3 6 3 3 . . . . 3 . . . 3 3 3 . . 3 . . 1 1 1 .

Also, for all m and all n:

1 = ∧ n 1 = △ n 0 ⊂ △ n m − 1 = ∧ n m {\displaystyle 1=\wedge _{n}^{1}=\vartriangle _{n}^{0}\subset \vartriangle _{n}^{m-1}=\wedge _{n}^{m}}

Number of coefficients

For the nth component ((m − 1)-simplex) of Pascal's m-simplex, the number of the coefficients of multinomial expansion it consists of is given by:

( ( n − 1 ) + ( m − 1 ) ( m − 1 ) ) + ( n + ( m − 2 ) ( m − 2 ) ) = ( n + ( m − 1 ) ( m − 1 ) ) = ( ( m n ) ) , {\displaystyle {(n-1)+(m-1) \choose (m-1)}+{n+(m-2) \choose (m-2)}={n+(m-1) \choose (m-1)}=\left(\!\!{\binom {m}{n}}\!\!\right),}

(where the latter is the multichoose notation). We can see this either as a sum of the number of coefficients of an (n − 1)th component ((m − 1)-simplex) of Pascal's m-simplex with the number of coefficients of an nth component ((m − 2)-simplex) of Pascal's (m − 1)-simplex, or by a number of all possible partitions of an nth power among m exponents.

Example

Number of coefficients of nth component ((m − 1)-simplex) of Pascal's m-simplex

m-simplex	nth component	n = 0	n = 1	n = 2	n = 3	n = 4	n = 5
1-simplex	0-simplex	1	1	1	1	1	1
2-simplex	1-simplex	1	2	3	4	5	6
3-simplex	2-simplex	1	3	6	10	15	21
4-simplex	3-simplex	1	4	10	20	35	56
5-simplex	4-simplex	1	5	15	35	70	126
6-simplex	5-simplex	1	6	21	56	126	252

The terms of this table comprise a Pascal triangle in the format of a symmetric Pascal matrix.

Symmetry

An nth component ((m − 1)-simplex) of Pascal's m-simplex has the (m!)-fold spatial symmetry.

Geometry

Orthogonal axes k1, ..., km in m-dimensional space, vertices of component at n on each axis, the tip at [0, ..., 0] for n = 0.

Numeric construction

Wrapped nth power of a big number gives instantly the nth component of a Pascal's simplex.

| b d p | n = ∑ | k | = n ( n k ) b d p ⋅ k ;     b , d ∈ N ,   n ∈ N 0 ,   k , p ∈ N 0 m ,   p :   p 1 = 0 , p i = ( n + 1 ) i − 2 {\displaystyle \left|b^{dp}\right|^{n}=\sum _{|k|=n}{{\binom {n}{k}}b^{dp\cdot k}};\ \ b,d\in \mathbb {N} ,\ n\in \mathbb {N} _{0},\ k,p\in \mathbb {N} _{0}^{m},\ p:\ p_{1}=0,p_{i}=(n+1)^{i-2}}

where b d p = ( b d p 1 , ⋯ , b d p m ) ∈ N m ,   p ⋅ k = ∑ i = 1 m p i k i ∈ N 0 {\displaystyle \textstyle b^{dp}=(b^{dp_{1}},\cdots ,b^{dp_{m}})\in \mathbb {N} ^{m},\ p\cdot k={\sum _{i=1}^{m}{p_{i}k_{i}}}\in \mathbb {N} _{0}} .