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http://functions.wolfram.com/04.21.23.0001.01
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Sum[\[Ellipsis] Sum[Signature[{Subscript[\[Tau], 1], Subscript[\[Tau], 2],
\[Ellipsis], Subscript[\[Tau], r], Subscript[\[Nu], r + 1],
\[Ellipsis], Subscript[\[Nu], n]}]
Signature[{Subscript[\[Tau], 1], Subscript[\[Tau], 2], \[Ellipsis],
Subscript[\[Tau], r], Subscript[\[Mu], r + 1], \[Ellipsis],
Subscript[\[Mu], n]}], {Subscript[\[Tau], r], 1, n}],
{Subscript[\[Tau], 1], 1, n}, {Subscript[\[Tau], 2], 1, n}] ==
r! Sum[\[Ellipsis] Sum[Signature[{Subscript[\[Mu], r + 1], \[Ellipsis],
Subscript[\[Mu], n]}] Product[KroneckerDelta[Subscript[\[Nu], k],
Subscript[\[Mu], k]], {k, r + 1, n}], {Subscript[\[Mu], n], 1, n}],
{Subscript[\[Mu], r + 1], 1, n}, {Subscript[\[Mu], r + 2], 1, n}]
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Tau]", "1"], "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Tau]", "2"], "=", "1"]], "n"], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Tau]", "r"], "=", "1"]], "n"], RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Tau]", "1"], ",", SubscriptBox["\[Tau]", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["\[Tau]", "r"], ",", SubscriptBox["\[Nu]", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["\[Nu]", "n"]]], "}"]], "]"]], RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Tau]", "1"], ",", SubscriptBox["\[Tau]", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["\[Tau]", "r"], ",", SubscriptBox["\[Mu]", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["\[Mu]", "n"]]], "}"]], "]"]]]]]]]]]]]], " ", "\[Equal]", RowBox[List[RowBox[List["r", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Mu]", RowBox[List["r", "+", "1"]]], "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Mu]", RowBox[List["r", "+", "2"]]], "=", "1"]], "n"], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Mu]", "n"], "=", "1"]], "n"], RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Mu]", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["\[Mu]", "n"]]], "}"]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", RowBox[List["r", "+", "1"]]]], "n"], RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["\[Nu]", "k"], ",", SubscriptBox["\[Mu]", "k"]]], "]"]]]]]]]]]]]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> τ </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> τ </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mo> … </mo> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> τ </mi> <mi> r </mi> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> ε </mi> <mrow> <msub> <mi> τ </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> τ </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> τ </mi> <mi> r </mi> </msub> <mo> , </mo> <msub> <mi> ν </mi> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> ν </mi> <mi> n </mi> </msub> </mrow> </msub> <mo> ⁢ </mo> <msub> <mi> ε </mi> <mrow> <msub> <mi> τ </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> τ </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> τ </mi> <mi> r </mi> </msub> <mo> , </mo> <msub> <mi> μ </mi> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> μ </mi> <mi> n </mi> </msub> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> r </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> μ </mi> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> μ </mi> <mrow> <mi> r </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mo> … </mo> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <msub> <mi> μ </mi> <mi> n </mi> </msub> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> ε </mi> <mrow> <msub> <mi> μ </mi> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> μ </mi> <mi> n </mi> </msub> </mrow> </msub> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> n </mi> </munderover> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> ν </mi> <mi> k </mi> </msub> <mo> , </mo> <msub> <mi> μ </mi> <mi> k </mi> </msub> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> τ </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> τ </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <ci> … </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> τ </ci> <ci> r </ci> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Signature </ci> <list> <apply> <ci> Subscript </ci> <ci> τ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> τ </ci> <cn type='integer'> 2 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> τ </ci> <ci> r </ci> </apply> <apply> <ci> Subscript </ci> <ci> ν </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> ν </ci> <ci> n </ci> </apply> </list> </apply> <apply> <ci> Signature </ci> <list> <apply> <ci> Subscript </ci> <ci> τ </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> τ </ci> <cn type='integer'> 2 </cn> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> τ </ci> <ci> r </ci> </apply> <apply> <ci> Subscript </ci> <ci> μ </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> μ </ci> <ci> n </ci> </apply> </list> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <factorial /> <ci> r </ci> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> μ </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> 2 </cn> </apply> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> μ </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <ci> … </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> μ </ci> <ci> n </ci> </apply> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Signature </ci> <list> <apply> <ci> Subscript </ci> <ci> μ </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> … </ci> <apply> <ci> Subscript </ci> <ci> μ </ci> <ci> n </ci> </apply> </list> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <plus /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> KroneckerDelta </ci> <apply> <ci> Subscript </ci> <ci> ν </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> μ </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Tau]_", "1"], "=", "1"]], "n_"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Tau]_", "2"], "=", "1"]], "n_"], RowBox[List["\[Ellipsis]_", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Tau]_", "r_"], "=", "1"]], "n_"], RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Tau]_", "1"], ",", SubscriptBox["\[Tau]_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["\[Tau]_", "r_"], ",", SubscriptBox["\[Nu]_", RowBox[List["r_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["\[Nu]_", "n_"]]], "}"]], "]"]], " ", RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Tau]_", "1"], ",", SubscriptBox["\[Tau]_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["\[Tau]_", "r_"], ",", SubscriptBox["\[Mu]_", RowBox[List["r_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["\[Mu]_", "n_"]]], "}"]], "]"]]]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["r", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Mu]", RowBox[List["r", "+", "1"]]], "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Mu]", RowBox[List["r", "+", "2"]]], "=", "1"]], "n"], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Mu]\[Mu]", "n"], "=", "1"]], "n"], RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["\[Mu]", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["\[Mu]\[Mu]", "n"]]], "}"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", RowBox[List["r", "+", "1"]]]], "n"], RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["\[Nu]\[Nu]", "k"], ",", SubscriptBox["\[Mu]\[Mu]", "k"]]], "]"]]]]]]]]]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
