Combinatorial Analysis - 26.15 Permutations: Matrix Notation

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
26.15.E3 R ⁒ ( x , B ) = βˆ‘ j = 0 n r j ⁒ ( B ) ⁒ x j 𝑅 π‘₯ 𝐡 superscript subscript 𝑗 0 𝑛 subscript π‘Ÿ 𝑗 𝐡 superscript π‘₯ 𝑗 {\displaystyle{\displaystyle R(x,B)=\sum_{j=0}^{n}r_{j}(B)\,x^{j}}}
R(x,B) = \sum_{j=0}^{n}r_{j}(B)\,x^{j}		 

R(x , B) = sum(r[j](B)* (x)^(j), j = 0..n)
 
R[x , B] == Sum[Subscript[r, j][B]* (x)^(j), {j, 0, n}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
26.15.E4 R ⁒ ( x , B ) = R ⁒ ( x , B 1 ) ⁒ R ⁒ ( x , B 2 ) 𝑅 π‘₯ 𝐡 𝑅 π‘₯ subscript 𝐡 1 𝑅 π‘₯ subscript 𝐡 2 {\displaystyle{\displaystyle R(x,B)=R(x,B_{1})\,R(x,B_{2})}}
R(x,B) = R(x,B_{1})\,R(x,B_{2})		 

R(x , B) = R(x , B[1])* R(x , B[2])
 
R[x , B] == R[x , Subscript[B, 1]]* R[x , Subscript[B, 2]]
 Skipped - no semantic math Skipped - no semantic math - -
26.15.E6 N ⁒ ( x , B ) = βˆ‘ k = 0 n N k ⁒ ( B ) ⁒ x k 𝑁 π‘₯ 𝐡 superscript subscript π‘˜ 0 𝑛 subscript 𝑁 π‘˜ 𝐡 superscript π‘₯ π‘˜ {\displaystyle{\displaystyle N(x,B)=\sum_{k=0}^{n}N_{k}(B)\,x^{k}}}
N(x,B) = \sum_{k=0}^{n}N_{k}(B)\,x^{k}		 

N(x , B) = sum(N[k](B)* (x)^(k), k = 0..n)
 
N[x , B] == Sum[Subscript[N, k][B]* (x)^(k), {k, 0, n}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
26.15.E7 N ⁒ ( x , B ) = βˆ‘ k = 0 n r k ⁒ ( B ) ⁒ ( n - k ) ! ⁒ ( x - 1 ) k 𝑁 π‘₯ 𝐡 superscript subscript π‘˜ 0 𝑛 subscript π‘Ÿ π‘˜ 𝐡 𝑛 π‘˜ superscript π‘₯ 1 π‘˜ {\displaystyle{\displaystyle N(x,B)=\sum_{k=0}^{n}r_{k}(B)(n-k)!(x-1)^{k}}}
N(x,B) = \sum_{k=0}^{n}r_{k}(B)(n-k)!(x-1)^{k}		 

N(x , B) = sum(r[k](B)*factorial(n - k)*(x - 1)^(k), k = 0..n)
 
N[x , B] == Sum[Subscript[r, k][B]*(n - k)!*(x - 1)^(k), {k, 0, n}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
26.15.E8 N 0 ⁒ ( B ) ≑ N ⁒ ( 0 , B ) = βˆ‘ k = 0 n ( - 1 ) k ⁒ r k ⁒ ( B ) ⁒ ( n - k ) ! equal-by definition subscript 𝑁 0 𝐡 𝑁 0 𝐡 superscript subscript π‘˜ 0 𝑛 superscript 1 π‘˜ subscript π‘Ÿ π‘˜ 𝐡 𝑛 π‘˜ {\displaystyle{\displaystyle N_{0}(B)\equiv N(0,B)=\sum_{k=0}^{n}(-1)^{k}r_{k}% (B)(n-k)!}}
N_{0}(B)\defeq N(0,B) = \sum_{k=0}^{n}(-1)^{k}r_{k}(B)(n-k)!		 

N[0](B) = N(0 , B) = sum((- 1)^(k)* r[k](B)*factorial(n - k), k = 0..n)

Subscript[N, 0][B] == N[0 , B] == Sum[(- 1)^(k)* Subscript[r, k][B]*(n - k)!, {k, 0, n}, GenerateConditions->None]
Failure Failure Error Error
26.15.E9 r k ⁒ ( B ) = 2 ⁒ n 2 ⁒ n - k ⁒ ( 2 ⁒ n - k k ) subscript π‘Ÿ π‘˜ 𝐡 2 𝑛 2 𝑛 π‘˜ binomial 2 𝑛 π‘˜ π‘˜ {\displaystyle{\displaystyle r_{k}(B)=\frac{2n}{2n-k}\genfrac{(}{)}{0.0pt}{}{2% n-k}{k}}}
r_{k}(B) = \frac{2n}{2n-k}\binom{2n-k}{k}		 

r[k](B) = (2*n)/(2*n - k)*binomial(2*n - k,k)

Subscript[r, k][B] == Divide[2*n,2*n - k]*Binomial[2*n - k,k]
Failure Failure
Failed [300 / 300]
Result: -1.500000000+.8660254040*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, r[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}


Result: -3.500000000+.8660254040*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, r[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.5, 0.8660254037844386]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[n, 1], Rule[Subscript[r, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.5, 0.8660254037844386]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[n, 2], Rule[Subscript[r, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
26.15.E10 2 ⁒ ( n ! ) ⁒ N 0 ⁒ ( B ) = 2 ⁒ ( n ! ) ⁒ βˆ‘ k = 0 n ( - 1 ) k ⁒ 2 ⁒ n 2 ⁒ n - k ⁒ ( 2 ⁒ n - k k ) ⁒ ( n - k ) ! 2 𝑛 subscript 𝑁 0 𝐡 2 𝑛 superscript subscript π‘˜ 0 𝑛 superscript 1 π‘˜ 2 𝑛 2 𝑛 π‘˜ binomial 2 𝑛 π‘˜ π‘˜ 𝑛 π‘˜ {\displaystyle{\displaystyle 2(n!)N_{0}(B)=2(n!)\sum_{k=0}^{n}(-1)^{k}\frac{2n% }{2n-k}\genfrac{(}{)}{0.0pt}{}{2n-k}{k}{(n-k)!}}}
2(n!)N_{0}(B) = 2(n!)\sum_{k=0}^{n}(-1)^{k}\frac{2n}{2n-k}\binom{2n-k}{k}{(n-k)!}		 

2*(factorial(n))*N[0](B) = 2*(factorial(n))*sum((- 1)^(k)*(2*n)/(2*n - k)*binomial(2*n - k,k)*factorial(n - k), k = 0..n)

2*((n)!)*Subscript[N, 0][B] == 2*((n)!)*Sum[(- 1)^(k)*Divide[2*n,2*n - k]*Binomial[2*n - k,k]*(n - k)!, {k, 0, n}, GenerateConditions->None]
Failure Failure
Failed [292 / 300]
Result: 3.000000001+1.732050808*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, N[0] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 2.000000002+3.464101616*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, N[0] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
 Skipped - Because timed out
26.15.E11 βˆ‘ k = 0 n r n - k ⁒ ( B ) ⁒ ( x - k + 1 ) k = ∏ j = 1 n ( x + b j - j + 1 ) superscript subscript π‘˜ 0 𝑛 subscript π‘Ÿ 𝑛 π‘˜ 𝐡 subscript π‘₯ π‘˜ 1 π‘˜ superscript subscript product 𝑗 1 𝑛 π‘₯ subscript 𝑏 𝑗 𝑗 1 {\displaystyle{\displaystyle\sum_{k=0}^{n}r_{n-k}(B)(x-k+1)_{k}=\prod_{j=1}^{n% }(x+b_{j}-j+1)}}
\sum_{k=0}^{n}r_{n-k}(B)(x-k+1)_{k} = \prod_{j=1}^{n}(x+b_{j}-j+1)		 

sum(r[n - k](B)*x - k + 1[k], k = 0..n) = product(x + b[j]- j + 1, j = 1..n)
 
Sum[Subscript[r, n - k][B]*Subscript[x - k + 1, k], {k, 0, n}, GenerateConditions->None] == Product[x + Subscript[b, j]- j + 1, {j, 1, n}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
26.15.E12 βˆ‘ k = 0 n r n - k ⁒ ( B ) ⁒ ( x - k + 1 ) k = x n superscript subscript π‘˜ 0 𝑛 subscript π‘Ÿ 𝑛 π‘˜ 𝐡 subscript π‘₯ π‘˜ 1 π‘˜ superscript π‘₯ 𝑛 {\displaystyle{\displaystyle\sum_{k=0}^{n}r_{n-k}(B)(x-k+1)_{k}=x^{n}}}
\sum_{k=0}^{n}r_{n-k}(B)(x-k+1)_{k} = x^{n}		 

sum(r[n - k](B)*x - k + 1[k], k = 0..n) = (x)^(n)
 
Sum[Subscript[r, n - k][B]*Subscript[x - k + 1, k], {k, 0, n}, GenerateConditions->None] == (x)^(n)
 Skipped - no semantic math Skipped - no semantic math - -
26.15.E13 r n - k ⁒ ( B ) = S ⁑ ( n , k ) subscript π‘Ÿ 𝑛 π‘˜ 𝐡 Stirling-number-second-kind-S 𝑛 π‘˜ {\displaystyle{\displaystyle r_{n-k}(B)=S\left(n,k\right)}}
r_{n-k}(B) = \StirlingnumberS@{n}{k}		 

r[n - k](B) = Stirling2(n, k)

Subscript[r, n - k][B] == StirlingS2[n, k]
Failure Failure
Failed [300 / 300]
Result: -.4999999996+.8660254040*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, r[n-k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}


Result: -.4999999996+.8660254040*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, r[n-k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.4999999999999999, 0.8660254037844386]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[n, 1], Rule[Subscript[r, Plus[Times[-1, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.4999999999999999, 0.8660254037844386]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[k, 1], Rule[n, 2], Rule[Subscript[r, Plus[Times[-1, k], n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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