Sign[StirlingS2[n, m]] == Piecewise[{{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> &#119982; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[ScriptCapitalS]&quot;, StirlingS2] </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> <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> StirlingS2 </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> <apply> <ci> Piecewise </ci> <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>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sign", "[", RowBox[List["StirlingS2", "[", RowBox[List["n_", ",", "m_"]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["\[Piecewise]", GridBox[List[List["1", RowBox[List[RowBox[List["1", "\[LessEqual]", "m", "\[LessEqual]", "n"]], "||", RowBox[List["m", "\[Equal]", "n", "\[Equal]", "0"]]]]], List["0", "True"]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]]]]]]

2007-05-02