Sign[StirlingS1[n, m]] == Piecewise[{{1, (1 <= m <= n && Element[(m + n)/2, Integers]) || m == n == 0}, {-1, 1 <= m <= n || m == n == 0}, {0, True}}]

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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> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox[&quot;S&quot;, StirlingS1] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mo> &#62305; </mo> <mtable> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> m </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8744; </mo> <mrow> <mi> m </mi> <mo> &#63449; </mo> <mi> n </mi> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> <mtd> <mrow> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> m </mi> <mo> &#8804; </mo> <mi> n </mi> </mrow> <mo> &#8744; </mo> <mrow> <mi> m </mi> <mo> &#63449; </mo> <mi> n </mi> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mi> True </mi> </mtd> </mtr> </mtable> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <ci> StirlingS1 </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <ci> Piecewise </ci> <list> <list> <cn type='integer'> 1 </cn> <apply> <or /> <apply> <and /> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> m </ci> <ci> n </ci> </apply> <apply> <in /> <apply> <times /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </apply> <apply> <eq /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </list> <list> <cn type='integer'> -1 </cn> <apply> <or /> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> m </ci> <ci> n </ci> </apply> <apply> <eq /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </list> <list> <cn type='integer'> 0 </cn> <true /> </list> </list> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02