Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











StirlingS2






Mathematica Notation

Traditional Notation









Integer Functions > StirlingS2[n,m] > Specific values > Specialized values > For fixed m





http://functions.wolfram.com/04.15.03.0046.01









  


  










Input Form





StirlingS2[10, m] == ((-1)^(m - 1)/(m - 1)!) (KroneckerDelta[m - 1] + 511 (1 - m) KroneckerDelta[m - 2] + 9330 (-2 + m) (-1 + m) KroneckerDelta[-3 + m] - 34105 (-3 + m) (-2 + m) (-1 + m) KroneckerDelta[-4 + m] + 42525 Pochhammer[m - 4, 4] KroneckerDelta[-5 + m] - 22827 Pochhammer[m - 5, 5] KroneckerDelta[-6 + m] + 5880 Pochhammer[m - 6, 6] KroneckerDelta[-7 + m] - 750 Pochhammer[m - 7, 7] KroneckerDelta[-8 + m] + 45 Pochhammer[m - 8, 8] KroneckerDelta[-9 + m] - Pochhammer[m - 9, 9] KroneckerDelta[-10 + m]) /; Element[m, Integers] && m >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StirlingS2", "[", RowBox[List["10", ",", "m"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "1"]], "]"]], "+", RowBox[List["511", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "2"]], "]"]]]], "+", RowBox[List["9330", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "3"]], "+", "m"]], "]"]]]], "-", RowBox[List["34105", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "4"]], "+", "m"]], "]"]]]], " ", "+", RowBox[List["42525", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "4"]], ",", "4"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "5"]], "+", "m"]], "]"]]]], "-", RowBox[List["22827", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "5"]], ",", "5"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "6"]], "+", "m"]], "]"]]]], "+", RowBox[List["5880", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "6"]], ",", "6"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "7"]], "+", "m"]], "]"]]]], "-", RowBox[List["750", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "7"]], ",", "7"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "8"]], "+", "m"]], "]"]]]], "+", RowBox[List["45", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "8"]], ",", "8"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "9"]], "+", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "9"]], ",", "9"]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "10"]], "+", "m"]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msubsup> <semantics> <mi> &#119982; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[ScriptCapitalS]&quot;, StirlingS2] </annotation> </semantics> <mn> 10 </mn> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 511 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> </mrow> <mo> + </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> - </mo> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 9 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 9 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;9&quot;]], &quot;)&quot;]], &quot;9&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 10 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mn> 750 </mn> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 7 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;7&quot;]], &quot;)&quot;]], &quot;7&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 8 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mn> 22827 </mn> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 5 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;5&quot;]], &quot;)&quot;]], &quot;5&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 6 </mn> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mn> 34105 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 4 </mn> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <mn> 9330 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 3 </mn> </mrow> </msub> </mrow> <mo> + </mo> <mrow> <mn> 45 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 9 </mn> </mrow> </msub> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 8 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;8&quot;]], &quot;)&quot;]], &quot;8&quot;], Pochhammer] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mn> 5880 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 7 </mn> </mrow> </msub> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 6 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;6&quot;]], &quot;)&quot;]], &quot;6&quot;], Pochhammer] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mn> 42525 </mn> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> m </mi> <mo> - </mo> <mn> 5 </mn> </mrow> </msub> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;m&quot;, &quot;-&quot;, &quot;4&quot;]], &quot;)&quot;]], &quot;4&quot;], Pochhammer] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> StirlingS2 </ci> <cn type='integer'> 10 </cn> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 511 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -9 </cn> </apply> <cn type='integer'> 9 </cn> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -10 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 750 </cn> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -7 </cn> </apply> <cn type='integer'> 7 </cn> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -8 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 22827 </cn> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -5 </cn> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -6 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 34105 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -3 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -4 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 9330 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -9 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -8 </cn> </apply> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 5880 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -7 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -6 </cn> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 42525 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -5 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -4 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StirlingS2", "[", RowBox[List["10", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "1"]], "]"]], "+", RowBox[List["511", " ", RowBox[List["(", RowBox[List["1", "-", "m"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["m", "-", "2"]], "]"]]]], "+", RowBox[List["9330", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "3"]], "+", "m"]], "]"]]]], "-", RowBox[List["34105", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "m"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "4"]], "+", "m"]], "]"]]]], "+", RowBox[List["42525", " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "4"]], ",", "4"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "5"]], "+", "m"]], "]"]]]], "-", RowBox[List["22827", " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "5"]], ",", "5"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "6"]], "+", "m"]], "]"]]]], "+", RowBox[List["5880", " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "6"]], ",", "6"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "7"]], "+", "m"]], "]"]]]], "-", RowBox[List["750", " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "7"]], ",", "7"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "8"]], "+", "m"]], "]"]]]], "+", RowBox[List["45", " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "8"]], ",", "8"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "9"]], "+", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["m", "-", "9"]], ",", "9"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["-", "10"]], "+", "m"]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.