Pascal's simplex

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In mathematics, Pascal's simplex is a generalisation of Pascal's triangle into arbitrary number of dimensions, based on the multinomial theorem.

Generic Pascal's m-simplex[edit]

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 \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 \wedge^m_n denote its nth component, itself a finite (m − 1)-simplex with the edge length n, with a notational equivalent \vartriangle^{m-1}_n.

nth component[edit]

\wedge^m_n = \vartriangle^{m-1}_n consists of the coefficients of multinomial expansion of a polynomial with m terms raised to the power of n:

|x|^n=\sum_{|k|=n}{\binom{n}{k}x^k};\ \ x\in\mathbb{R}^m,\ k\in\mathbb{N}^m_0,\ n\in\mathbb{N}_0,\ m\in\mathbb{N}

where \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 \wedge^4[edit]

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

First four components of Pascal's 4-simplex.

Specific Pascal's simplices[edit]

Pascal's 1-simplex[edit]

\wedge^1 is not known by any special name.

First four components of Pascal's line.

nth component[edit]

\wedge^1_n = \vartriangle^0_n (a point) is the coefficient of multinomial expansion of a polynomial with 1 term raised to the power of n:

(x_1)^n = \sum_{k_1=n} {n \choose k_1} x_1^{k_1};\ \ k_1, n \in \mathbb{N}_0
Arrangement of \vartriangle^0_n[edit]
\textstyle {n \choose n}

which equals 1 for all n.

Pascal's 2-simplex[edit]

\wedge^2 is known as Pascal's triangle (sequence A007318 in OEIS).

First four components of Pascal's triangle.

nth component[edit]

\wedge^2_n = \vartriangle^1_n (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 = \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 \vartriangle^1_n[edit]
\textstyle {n \choose n, 0}, {n \choose n - 1, 1}, \cdots, {n \choose 1, n - 1}, {n \choose 0, n}

Pascal's 3-simplex[edit]

\wedge^3 is known as Pascal's tetrahedron (sequence A046816 in OEIS).

First four components of Pascal's tetrahedron.

nth component[edit]

\wedge^3_n = \vartriangle^2_n (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 = \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 \vartriangle^2_n[edit]

\begin{align}
\textstyle {n \choose n, 0, 0} &, \textstyle {n \choose n - 1, 1, 0}, \cdots\cdots, {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}, \cdots\cdots, {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{align}

Properties[edit]

Inheritance of components[edit]

\wedge^m_n = \vartriangle^{m-1}_n is numerically equal to each (m − 1)-face (there is m + 1 of them) of \vartriangle^m_n = \wedge^{m+1}_n, or:

\wedge^m_n = \vartriangle^{m-1}_n \subset\ \vartriangle^m_n = \wedge^{m+1}_n

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

\wedge^m \subset \wedge^{m+1}

Example[edit]

        \wedge^1         \wedge^2        \wedge^3         \wedge^4

\wedge^m_0     1          1          1          1

\wedge^m_1     1         1 1        1 1        1 1  1
                              1          1

\wedge^m_2     1        1 2 1      1 2 1      1 2 1  2 2  1
                             2 2        2 2    2
                              1          1

\wedge^m_3     1       1 3 3 1    1 3 3 1    1 3 3 1  3 6 3  3 3  1
                            3 6 3      3 6 3    6 6    3
                             3 3        3 3      3
                              1          1

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

Equality of sub-faces[edit]

Conversely, \wedge^{m+1}_n = \vartriangle^m_n is (m + 1)-times bounded by \vartriangle^{m-1}_n = \wedge^m_n, or:

\wedge^{m+1}_n = \vartriangle^m_n \supset \vartriangle^{m-1}_n = \wedge^m_n

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

\wedge^{i+1}_n = \vartriangle^i_n \subset \vartriangle^{m>i}_n = \wedge^{m>i+1}_n

Example[edit]

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 = \wedge^1_n = \vartriangle^0_n \subset \vartriangle^{m-1}_n = \wedge^m_n

Number of coefficients[edit]

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) \choose (m-1)} + {n + (m - 2) \choose (m - 2)} = {n + (m - 1) \choose (m - 1)},

that is, either by 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[edit]

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

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

Symmetry[edit]

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

Geometry[edit]

(Orthogonal axes k_1 ... k_m in m-dimensional space, vertices of component at n on each axe, the tip at [0,...,0] for n=0.)

Numeric construction[edit]

(Wrapped n-th power of a big number gives instantly the n-th component of a Pascal's simplex.)

\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 \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.