Involutions[n_]:=(I/Sqrt[2])^nHermiteH[n,-I/Sqrt[2]];
Involutions[n_]:=Sum[StirlingS1[n,k]*2^k*BellB[k,1/2],{k,0,n}]
In[]:=
Involutions[4]
In[]:=
10
Out[]=
Array[Involutions,20]
In[]:=
{1,2,4,10,26,76,232,764,2620,9496,35696,140152,568504,2390480,10349536,46206736,211799312,997313824,4809701440,23758664096}
Out[]=
self-inverse permutations

All total orderings

IntegerPartitions[6]
In[]:=
{{6},{5,1},{4,2},{4,1,1},{3,3},{3,2,1},{3,1,1,1},{2,2,2},{2,2,1,1},{2,1,1,1,1},{1,1,1,1,1,1}}
Out[]=
youngall=Catenate[Tableaux/@IntegerPartitions[6]]
In[]:=
Out[]=
Length[young]
In[]:=
76
Out[]=
Divisors[76]
In[]:=
{1,2,4,19,38,76}
Out[]=
skewall=SkewPoset/@YoungTableauToPoset/@Catenate[Tableaux/@IntegerPartitions[6]];
In[]:=
SkewPosetDiagram[#,ImageSize30]&/@skewall
In[]:=
Out[]=
Grid[Partition[Column[#,AlignmentCenter]&/@Transpose[{Grid[Map[Text[Style[#,Small]]&,#,{2}],FrameAll,FrameStyleLightGray]&/@young,SkewPosetDiagram[#,ImageSize70]&/@skew}],8],FrameAll]
In[]:=
Out[]=
n
2
∑
k=0
(2k-1)!!
n
2k

In[]:=
1-n

1
2
(-1+n)
2
HypergeometricU
1-n
2
,
3
2
,-
1
2

Out[]=
Series[%63,{n,Infinity,2}]
In[]:=
-
π
2
+
Log[2]
2
n+
π
2
-
Log[2]
2
+
3
O
1
n


HypergeometricU
1
2
-
n
2
,
3
2
,-
1
2

Out[]=
DiscreteAsymptotic[%,nInfinity]
In[]:=
π
2
+n-
π
2
+
Log[2]
2

HypergeometricU
1
2
-
n
2
,
3
2
,-
1
2

2
Out[]=
Involutions[n_]:=Sum[(2k-1)!!Binomial[n,2k],{k,0,n/2}];
In[]:=
as[n_]:=Exp[Sqrt[n]-n/2-1/4]n^(n/2)/Sqrt[2]
In[]:=
RecurrenceTable[{(-1-n)a[n]-a[1+n]+a[2+n]0,a[1]1,a[2]2},a,{n,200}]
In[]:=
Out[]=
%/Array[Factorial,200]//N
In[]:=
Out[]=
Table[N[
n/2

-n/2
n
],{n,200}]
In[]:=
Out[]=
%85/%93
In[]:=
Out[]=
ListLinePlot[Differences[%94]]
In[]:=
Out[]=
Table[N[n^-(n/2)],{n,200}]
In[]:=
Out[]=
Array[as,20]
In[]:=
Out[]=
N[%]
In[]:=
{0.907943,1.6666,3.60887,8.81113,23.6435,68.5933,212.696,698.978,2418.58,8765.94,33139.3,130218.,530274.,2.23217×
6
10
,9.69176×
6
10
,4.33205×
7
10
,1.99006×
8
10
,9.38143×
8
10
,4.5323×
9
10
,2.24125×
10
10
}
Out[]=
%/%68
In[]:=
{0.907943,0.833301,0.902218,0.881113,0.909366,0.902543,0.916792,0.914893,0.923123,0.92312,0.928376,0.929122,0.932753,0.933775,0.936444,0.937536,0.939597,0.94067,0.942325,0.94334}
Out[]=
DiscreteAsymptotic[Exp[Sqrt[n]-n/2-1/4]n^(n/2)/(Sqrt[2]n!),nInfinity]
In[]:=
-
1
4
+
n
+
1
2
n(1-Log[n])

2
n
π
Out[]=
FullSimplify[%]
In[]:=
-
1
4
+
n
+
1
2
n(-1+Log[n])

2
Out[]=
DiscreteAsymptotic[n!,nInfinity]
In[]:=
-n

1
2
+n
n
2π
Out[]=
DiscreteAsymptotic[Exp[Sqrt[n]-n/2-1/4]n^(n/2)/(Sqrt[2]n!),nInfinity]
In[]:=
-
1
4
+
n
+
1
2
n(1-Log[n])

2
n
π
Out[]=
Plot-
1
4
+
n
+
1
2
n(1-Log[n]),{n,0,100}
In[]:=
Out[]=
Plot
1
2
n(0-Log[n]),{n,0,100}
In[]:=
Out[]=
Plot
1
2
n(1-Log[n]),{n,0,100}
In[]:=
Out[]=
Show[%77,%]
In[]:=
Out[]=
Show[%,%%]
In[]:=
Out[]=
Exp[-1/2nLog[n]]
In[]:=
-n/2
n
Out[]=
Exp
1
2
n(1-Log[n])
In[]:=
1
2
n(1-Log[n])

Out[]=
FullSimplify[%]
In[]:=
-
1
2
n(-1+Log[n])

Out[]=
PowerExpand[%]
In[]:=
-
1
2
n(-1+Log[n])

Out[]=
n^-(n/2)Exp[n/2]
In[]:=
n/2

-n/2
n
Out[]=
-
n
2
n

In[]:=
n/2

-n/2
n
Out[]=
6!
In[]:=
720
Out[]=