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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle R(x,B) = R(x,B_{1})\,R(x,B_{2})}
R(x,B) = R(x,B_{1})\,R(x,B_{2})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle N_{0}(B)\defeq 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)!		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle r_{k}(B) = \frac{2n}{2n-k}\binom{2n-k}{k}}
r_{k}(B) = \frac{2n}{2n-k}\binom{2n-k}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 2(n!)N_{0}(B) = 2(n!)\sum_{k=0}^{n}(-1)^{k}\frac{2n}{2n-k}\binom{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)!}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle r_{n-k}(B) = \StirlingnumberS@{n}{k}}
r_{n-k}(B) = \StirlingnumberS@{n}{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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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