Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 LahL

Source Notebook

Evaluate the Lah number

Contributed by: Jan Mangaldan

ResourceFunction["LahL"][n,m]

gives the Lah number L(n,m).

Details

Integer mathematical function, suitable for both symbolic and numerical manipulation.
L(n,m) gives the number of ways of partitioning a set of n elements into m non-empty linearly ordered subsets.
ResourceFunction["LahL"] automatically threads over lists.

Examples

Basic Examples (1) 

Evaluate some Lah numbers:

In[1]:=
Table[ResourceFunction["LahL"][10, m], {m, 10}]
Out[1]=

Scope (1) 

LahL threads elementwise over lists:

In[2]:=
ResourceFunction["LahL"][{2, 4, 6}, 2]
Out[2]=

Applications (3) 

Plot Lah numbers on a logarithmic scale:

In[3]:=
ListPlot3D[ Table[Log[Abs[ResourceFunction["LahL"][n, m]] + 1], {n, 60}, {m, 60}]]
Out[3]=

Express Pochhammer as a linear combination of FactorialPower:

In[4]:=
With[{n = 7}, Pochhammer[x, n] == \!\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 1\), \(n\)]\(\* InterpretationBox[ TagBox[ FrameBox[ PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.65 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["\<\"LahL\"\>", "ResourceFunctionLabel", FontFamily->"Source Sans Pro"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->0.9 Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], Background->RGBColor[0.968627, 0.976471, 0.984314], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 1}}, FrameStyle->RGBColor[0.831373, 0.847059, 0.85098], RoundingRadius->4], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "\"LahL\""}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["LahL"], BoxID -> "LahL", Selectable->False][n, m] FactorialPower[x, m]\)\)] // FunctionExpand // Simplify
Out[4]=

Express FactorialPower as a linear combination of Pochhammer:

In[5]:=
With[{n = 7}, FactorialPower[x, n] == \!\( \*UnderoverscriptBox[\(\[Sum]\), \(m = 1\), \(n\)]\( \*SuperscriptBox[\((\(-1\))\), \(n - m\)] \* InterpretationBox[ TagBox[ FrameBox[ PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.65 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["\<\"LahL\"\>", "ResourceFunctionLabel", FontFamily->"Source Sans Pro"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->0.9 Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], Background->RGBColor[0.968627, 0.976471, 0.984314], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 1}}, FrameStyle->RGBColor[0.831373, 0.847059, 0.85098], RoundingRadius->4], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "\"LahL\""}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["LahL"], BoxID -> "LahL", Selectable->False][n, m] Pochhammer[x, m]\)\)] // FunctionExpand // Simplify
Out[5]=

Closed form of derivatives of ⅇ1/x:

In[6]:=
D[Exp[1/x], {x, 6}] // Factor
Out[6]=
In[7]:=
With[{n = 6}, (-1)^n Exp[1/x] \!\( \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(n\)]\* FractionBox[ RowBox[{ InterpretationBox[ TagBox[ FrameBox[ PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.65 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["\<\"LahL\"\>", "ResourceFunctionLabel", FontFamily->"Source Sans Pro"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->0.9 Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], Background->RGBColor[0.968627, 0.976471, 0.984314], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 1}}, FrameStyle->RGBColor[0.831373, 0.847059, 0.85098], RoundingRadius->4], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "\"LahL\""}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["LahL"], BoxID -> "LahL", Selectable->False], "[", RowBox[{"n", ",", "k"}], "]"}], SuperscriptBox["x", RowBox[{"n", "+", "k"}]]]\)] // Factor
Out[7]=

Properties and Relations (3) 

Generate values from the generating function:

In[8]:=
Table[12!/m! SeriesCoefficient[Series[(t/(1 - t))^m, {t, 0, 12}], 12], {m, 5}]
Out[8]=
In[9]:=
Table[ResourceFunction["LahL"][12, m] , {m, 5}]
Out[9]=

Lah numbers can be expressed in terms of Stirling numbers of both kinds:

In[10]:=
Table[\!\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 0\), \(n\)]\( \*SuperscriptBox[\((\(-1\))\), \(n - j\)] StirlingS1[n, j] StirlingS2[ j, m]\)\), {n, 5}, {m, n}]
Out[10]=
In[11]:=
Table[ResourceFunction["LahL"][n, m], {n, 5}, {m, n}]
Out[11]=

Lah numbers are given by a partial Bell polynomial with factorial arguments:

In[12]:=
Table[BellY[n, m, Table[k!, {k, n}]], {n, 5}, {m, n}]
Out[12]=
In[13]:=
Table[ResourceFunction["LahL"][n, m], {n, 5}, {m, n}]
Out[13]=

Version History

  • 1.0.0 – 17 May 2021

Source Metadata

  • Citation:
    • Daboul, S., Mangaldan, J., Spivey, M. Z., Taylor, P.J., "The Lah Numbers and the nth Derivative of e^(1/x)." Mathematics Magazine, vol. 86, no. 1, 39–47, 2013. DOI: 10.4169/math.mag.86.1.039

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