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Signature




Mathematica Notation

Traditional Notation









Integer Functions > Signature[{n1,n2,...,nd}] > Transformations > Products, sums, and powers of the direct function > Products of the direct function





http://functions.wolfram.com/04.21.16.0002.01









  


  








Input Form





Signature[{Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, r - 1], Subscript[n, r], Subscript[n, r + 1], \[Ellipsis], Subscript[n, d]}] Signature[{Subscript[n, 1], Subscript[n, 2], \[Ellipsis], Subscript[n, r - 1], Subscript[m, r], Subscript[m, r + 1], \[Ellipsis], Subscript[m, d]}] == (-(((d - r)! r!)/d!)) Sum[Signature[{Subscript[m, r], Subscript[m, r + 1], \[Ellipsis], Subscript[m, d]}] Product[KroneckerDelta[Subscript[n, k], Subscript[m, k]], {k, r, d}], Permutations[Subscript[m, r], Subscript[m, r + 1], \[Ellipsis], Subscript[m, d]]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", RowBox[List["r", "-", "1"]]], ",", SubscriptBox["n", "r"], ",", SubscriptBox["n", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["n", "d"]]], "}"]], "]"]], RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["n", "1"], ",", SubscriptBox["n", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["n", RowBox[List["r", "-", "1"]]], ",", SubscriptBox["m", "r"], ",", SubscriptBox["m", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["m", "d"]]], "}"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["d", "-", "r"]], ")"]], "!"]], RowBox[List["r", "!"]]]], RowBox[List["d", "!"]]]]], RowBox[List["Sum", "[", RowBox[List[RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["m", "r"], ",", SubscriptBox["m", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["m", "d"]]], "}"]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "r"]], "d"], RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["n", "k"], ",", SubscriptBox["m", "k"]]], "]"]]]]]], ",", RowBox[List["Permutations", "[", RowBox[List[SubscriptBox["m", "r"], ",", SubscriptBox["m", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["m", "d"]]], "]"]]]], "]"]]]]]]]]








MathML Form







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</mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> r </mi> <mo> ! </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> d </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <munder> <mo> &#8721; </mo> <mrow> <mi> permutations </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> m </mi> <mi> r </mi> </msub> <mo> , </mo> <msub> <mi> m </mi> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> m </mi> <mi> d </mi> </msub> </mrow> <mo> ) </mo> </mrow> </munder> <mtext> </mtext> <mrow> <msub> <mi> &#949; </mi> <mrow> <msub> <mi> m </mi> <mi> r </mi> </msub> <mo> , </mo> <msub> <mi> m </mi> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> m </mi> <mi> d </mi> </msub> </mrow> </msub> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mi> r </mi> </mrow> <mi> d </mi> </munderover> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <msub> <mi> n </mi> <mi> k </mi> </msub> <mo> , </mo> <msub> <mi> m </mi> <mi> k </mi> </msub> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#949; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> 1 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> 2 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> r </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> d </ms> </apply> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> &#949; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> 1 </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> 2 </ms> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> r </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> d </ms> </apply> </list> </apply> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> d </ms> <ms> - </ms> <ms> r </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> ! </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> d </ms> <ms> ! </ms> </list> </apply> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8721; </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> permutations </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> r </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> d </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#949; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> r </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <apply> <ci> RowBox </ci> <list> <ms> r </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> , </ms> <ms> &#8230; </ms> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> d </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> r </ms> </list> </apply> <ms> d </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> InterpretationBox </ci> <ms> &#948; </ms> <ci> KroneckerDelta </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <false /> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> n </ms> <ms> k </ms> </apply> <ms> , </ms> <apply> <ci> SubscriptBox </ci> <ms> m </ms> <ms> k </ms> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", RowBox[List["r_", "-", "1"]]], ",", SubscriptBox["n_", "r_"], ",", SubscriptBox["n_", RowBox[List["r_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", "d_"]]], "}"]], "]"]], " ", RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["n_", "1"], ",", SubscriptBox["n_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["n_", RowBox[List["r_", "-", "1"]]], ",", SubscriptBox["m_", "r_"], ",", SubscriptBox["m_", RowBox[List["r_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["m_", "d_"]]], "}"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["d", "-", "r"]], ")"]], "!"]], " ", RowBox[List["r", "!"]]]], ")"]], " ", RowBox[List["Sum", "[", RowBox[List[RowBox[List[RowBox[List["Signature", "[", RowBox[List["{", RowBox[List[SubscriptBox["mm", "r"], ",", SubscriptBox["m", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["mm", "d"]]], "}"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "r"]], "d"], RowBox[List["KroneckerDelta", "[", RowBox[List[SubscriptBox["nn", "k"], ",", SubscriptBox["mm", "k"]]], "]"]]]]]], ",", RowBox[List["Permutations", "[", RowBox[List[SubscriptBox["mm", "r"], ",", SubscriptBox["m", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["mm", "d"]]], "]"]]]], "]"]]]], RowBox[List["d", "!"]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29