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Pascal's simplex
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<h2 id="generic-pascals-m-simplex">Generic Pascal's <i>m</i>-simplex</h2> <p>Let <i>m</i> (<i>m</i> > 0) be a number of terms of a polynomial and <i>n</i> (<i>n</i> ≥ 0) be a power the polynomial is raised to. </p><p>Let ∧ {\displaystyle \wedge } <i>m</i> denote a Pascal's <i>m</i>-<a href="/facts/Simplex/0FCfNRN8">simplex</a>. Each Pascal's <i>m</i>-<a href="/facts/Simplex/0FCfNRN8">simplex</a> is a <a href="/facts/Semi-infinite/0c13Cg1u">semi-infinite</a> object, which consists of an infinite series of its components. </p><p>Let ∧ {\displaystyle \wedge } <i>m</i><i>n</i> denote its <i>n</i>th component, itself a finite (<i>m</i> − 1)-<a href="/facts/Simplex/0FCfNRN8">simplex</a> with the edge length <i>n</i>, with a notational equivalent △ n m − 1 {\displaystyle \vartriangle _{n}^{m-1}} . </p> <h3><i>n</i>th component</h3> <p> ∧ n m = △ n m − 1 {\displaystyle \wedge _{n}^{m}=\vartriangle _{n}^{m-1}} consists of the <a href="/facts/Multinomial_theorem/0kyhtHvf">coefficients of multinomial expansion</a> of a polynomial with <i>m</i> terms raised to the power of <i>n</i>: </p> | 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} } <p>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}}}.} </p> <h3>Example for ⋀4</h3> <p>Pascal's 4-simplex (sequence A189225 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>), sliced along the <i>k</i>4. All points of the same color belong to the same <i>n</i>th component, from red (for <i>n</i> = 0) to blue (for <i>n</i> = 3). </p> <h2 id="specific-pascals-simplices">Specific Pascal's simplices</h2> <h3>Pascal's 1-simplex</h3> <p> ∧ {\displaystyle \wedge } 1 is not known by any special name. </p> <h4><i>n</i>th component</h4> <p> ∧ n 1 = △ n 0 {\displaystyle \wedge _{n}^{1}=\vartriangle _{n}^{0}} (a point) is the <a href="/facts/Multinomial_theorem/0kyhtHvf">coefficient of multinomial expansion</a> of a polynomial with 1 term raised to the power of <i>n</i>: </p> ( 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}} <h5>Arrangement of △ n 0 {\displaystyle \vartriangle _{n}^{0}} </h5> ( n n ) {\displaystyle \textstyle {n \choose n}} <p>which equals 1 for all <i>n</i>. </p> <h3>Pascal's 2-simplex</h3> <p> ∧ 2 {\displaystyle \wedge ^{2}} is known as <a href="/facts/Pascal%2527s_triangle/tb6BKXbc">Pascal's triangle</a> (sequence A007318 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>). </p> <h4><i>n</i>th component</h4> <p> ∧ n 2 = △ n 1 {\displaystyle \wedge _{n}^{2}=\vartriangle _{n}^{1}} (a line) consists of the coefficients of <a href="/facts/Binomial_expansion/hd4SmkUd">binomial expansion</a> of a polynomial with 2 terms raised to the power of <i>n</i>: </p> ( 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}} <h5>Arrangement of △ n 1 {\displaystyle \vartriangle _{n}^{1}} </h5> ( 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}} <h3>Pascal's 3-simplex</h3> <p> ∧ 3 {\displaystyle \wedge ^{3}} is known as <a href="/facts/Pascal%2527s_tetrahedron/2ukt5A26">Pascal's tetrahedron</a> (sequence A046816 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>). </p> <h4><i>n</i>th component</h4> <p> ∧ n 3 = △ n 2 {\displaystyle \wedge _{n}^{3}=\vartriangle _{n}^{2}} (a triangle) consists of the coefficients of <a href="/facts/Trinomial_expansion/xtlRHY8F">trinomial expansion</a> of a polynomial with 3 terms raised to the power of <i>n</i>: </p> ( 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}} <h5>Arrangement of △ n 2 {\displaystyle \vartriangle _{n}^{2}} </h5> ( 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}}} <h2 id="properties">Properties</h2> <h3>Inheritance of components</h3> <p> ∧ n m = △ n m − 1 {\displaystyle \wedge _{n}^{m}=\vartriangle _{n}^{m-1}} is numerically equal to each (<i>m</i> − 1)-face (there is <i>m</i> + 1 of them) of △ n m = ∧ n m + 1 {\displaystyle \vartriangle _{n}^{m}=\wedge _{n}^{m+1}} , or: </p> ∧ 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}} <p>From this follows, that the whole ∧ m {\displaystyle \wedge ^{m}} is (<i>m</i> + 1)-times included in ∧ m + 1 {\displaystyle \wedge ^{m+1}} , or: </p> ∧ m ⊂ ∧ m + 1 {\displaystyle \wedge ^{m}\subset \wedge ^{m+1}} <h4>Example</h4> <table><tbody><tr><th></th><th> ∧ 1 {\displaystyle \wedge ^{1}} </th><th> ∧ 2 {\displaystyle \wedge ^{2}} </th><th> ∧ 3 {\displaystyle \wedge ^{3}} </th><th> ∧ 4 {\displaystyle \wedge ^{4}} </th></tr><tr><th> ∧ 0 m {\displaystyle \wedge _{0}^{m}} </th><td> 1 </td><td> 1</td><td> 1</td><td> 1</td></tr><tr><th> ∧ 1 m {\displaystyle \wedge _{1}^{m}} </th><td> 1 </td><td> 1 1</td><td> 1 1 1</td><td> 1 1 1 1</td></tr><tr><th> ∧ 2 m {\displaystyle \wedge _{2}^{m}} </th><td> 1 </td><td> 1 2 1</td><td> 1 2 1 2 2 1</td><td> 1 2 1 2 2 1 2 2 2 1</td></tr><tr><th> ∧ 3 m {\displaystyle \wedge _{3}^{m}} </th><td> 1 </td><td>1 3 3 1</td><td>1 3 3 1 3 6 3 3 3 1</td><td>1 3 3 1 3 6 3 3 3 1 3 6 3 6 6 3 3 3 3 1</td></tr></tbody></table> <p>For more terms in the above array refer to (sequence A191358 in the <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>) </p> <h3>Equality of sub-faces</h3> <p>Conversely, ∧ n m + 1 = △ n m {\displaystyle \wedge _{n}^{m+1}=\vartriangle _{n}^{m}} is (<i>m</i> + 1)-times bounded by △ n m − 1 = ∧ n m {\displaystyle \vartriangle _{n}^{m-1}=\wedge _{n}^{m}} , or: </p> ∧ 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}} <p>From this follows, that for given <i>n</i>, all <i>i</i>-faces are numerically equal in <i>n</i>th components of all Pascal's (<i>m</i> > <i>i</i>)-simplices, or: </p> ∧ 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}} <h4>Example</h4> <p>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): </p> 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 . <p>Also, for all <i>m</i> and all <i>n</i>: </p> 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}} <h3>Number of coefficients</h3> <p>For the <i>n</i>th component ((<i>m</i> − 1)-simplex) of Pascal's <i>m</i>-simplex, the number of the <a href="/facts/Multinomial_theorem/0kyhtHvf">coefficients of multinomial expansion</a> it consists of is given by: </p> ( ( 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),} <p>(where the latter is the <a href="/facts/Multiset/bPFJGMDo">multichoose</a> notation). We can see this either as a sum of the number of coefficients of an (<i>n</i> − 1)th component ((<i>m</i> − 1)-simplex) of Pascal's <i>m</i>-simplex with the number of coefficients of an <i>n</i>th component ((<i>m</i> − 2)-simplex) of Pascal's (<i>m</i> − 1)-simplex, or by a number of all possible partitions of an <i>n</i>th power among <i>m</i> exponents. </p> <h4>Example</h4> Number of coefficients of <i>n</i>th component ((<i>m</i> − 1)-simplex) of Pascal's <i>m</i>-simplex<table><tbody><tr><th>m-simplex</th><th><i>n</i>th component</th><th><i>n</i> = 0</th><th><i>n</i> = 1</th><th><i>n</i> = 2</th><th><i>n</i> = 3</th><th><i>n</i> = 4</th><th><i>n</i> = 5</th></tr><tr><td>1-simplex</td><th>0-simplex</th><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>2-simplex</td><th>1-simplex</th><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td></tr><tr><td>3-simplex</td><th>2-simplex</th><td>1</td><td>3</td><td>6</td><td>10</td><td>15</td><td>21</td></tr><tr><td>4-simplex</td><th>3-simplex</th><td>1</td><td>4</td><td>10</td><td>20</td><td>35</td><td>56</td></tr><tr><td>5-simplex</td><th>4-simplex</th><td>1</td><td>5</td><td>15</td><td>35</td><td>70</td><td>126</td></tr><tr><td>6-simplex</td><th>5-simplex</th><td>1</td><td>6</td><td>21</td><td>56</td><td>126</td><td>252</td></tr></tbody></table> <p>The terms of this table comprise a Pascal triangle in the format of a symmetric <a href="/facts/Pascal_matrix/jYJuazpz">Pascal matrix</a>. </p> <h3>Symmetry</h3> <p>An <i>n</i>th component ((<i>m</i> − 1)-simplex) of Pascal's <i>m</i>-simplex has the (<i>m</i>!)-fold spatial symmetry. </p> <h3>Geometry</h3> <p>Orthogonal axes <i>k</i>1, ..., <i>k</i><i>m</i> in <i>m</i>-dimensional space, vertices of component at <i>n</i> on each axis, the tip at [0, ..., 0] for <i>n</i> = 0. </p> <h3>Numeric construction</h3> <p>Wrapped <i>n</i>th power of a big number gives instantly the <i>n</i>th component of a Pascal's simplex. </p> | 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}} <p>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}} . </p>