Stirling Number of the Second Kind

Author

Eric W. Weisstein
December 21, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/StirlingNumberoftheSecondKind.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Traditional Form

StirlingS2[n,m]//TraditionalForm

Table

StirlingS2[0,0]

ColumnForm[Table[StirlingS2[k,n],{k,0,6},{n,k}],Center]

Generating Function

V6

(∑_(n=1)^∞)S(n,k) (x^n)=((x^k)/((1-k x) ⋯ (1-4 x) (1-3 x) (1-2 x) (1-x)))

FullSimplify[(x^k)/FunctionExpand[Product[1-k x,{k,n}]],n∈Integers&&n>0&&x∈Reals]

With[{k=5},
Series[x^k/Times@@((1-#x)&/@Range[k]),{x,0,20}]
]

With[{k=5},Table[StirlingS2[n,k],{n,20}]]

Product[1-n x,{n,k}]

(((-x))^k) Pochhammer[((-1+x)/x),k]

Sum[StirlingS2[n,k](x^n),{n,∞}]

Sum[StirlingS2[n,k](x^n),{n,10}]

V7

Sum[StirlingS2[n,k](x^n),{n,k,∞}]

FullSimplify[%,k∈Integers]

GeneratingFunction[StirlingS2[n,k],n,x]

Sum[StirlingS2[n,k](x^n),{n,∞}]

SeriesCoefficient[((((-1))^k)/(Pochhammer[((-1+x)/x),k])),{x,0,n}]

Special Cases

(StirlingS2[n, 0])==(KroneckerDeltaSeq[n])

FunctionExpand[StirlingS2[n,0],n>0&&n∈Integers]

(StirlingS2[n, 1])==1

FunctionExpand[StirlingS2[n,1],n>0&&n∈Integers]

(StirlingS2[n, 2])==(2^(n-1))-1

FunctionExpand[StirlingS2[n,2],n>0&&n∈Integers]

SeriesCoefficient[((x^2)/((1-x)(1-2x))),{x,0,n}]

(StirlingS2[n, 3])==(1/6) (-3 (2^n)+(3^n)+3)

FunctionExpand[StirlingS2[n,3],n>0&&n∈Integers]//FullSimplify

SeriesCoefficient[((x^3)/((1-x)(1-2x)(1-3x))),{x,0,n}]

(StirlingS2[n, 4])==(1/24) (3 (2^(n+1))-4 (3^n)+(4^n)-4)

FunctionExpand[StirlingS2[n,4],n>0&&n∈Integers]//FullSimplify

FunctionExpand[StirlingS2[n,4],n>5&&n∈Integers]

Table[StirlingS2[n,4],{n,10}]

SeriesCoefficient[((x^4)/((1-x)(1-2x)(1-3x)(1-4x))),{x,0,n}]

(StirlingS2[n, n-1])==(Binomial[n, 2])

FunctionExpand[StirlingS2[n,n-1],n>0&&n∈Integers]

Sums

(1/(k!))(∑_(i=0)^(k-1))(((-1))^i) (Binomial[k, i]) (((k-i))^n)==(StirlingS2[n, k])

v4.2

Sum[(-1)^i Binomial[k,i](k-i)^n,{i,0,k-1}]/k!

v5.0

Sum[(-1)^i Binomial[k,i](k-i)^n,{i,0,k-1}]/k!

V7

Sum[(-1)^i Binomial[k,i](k-i)^n,{i,0,k-1}]/k!

(∑_(m=0)^n)(FactorialPower[x, m]) (StirlingS2[n, m])==(x^n)

V7

Sum[StirlingS2[n,m]FactorialPower[x,m],{m,0,n}]

Table[Sum[StirlingS2[n,m]FactorialPower[x,m],{m,0,n}],{n,10}]//FunctionExpand//Expand

V8

Sum[StirlingS2[n,m]FactorialPower[x,m],{m,0,n}]

(∑_(n=k)^∞)(((x^n) (StirlingS2[n, k]))/(n!))==(((((e^x)-1))^k)/(k!))

V7

Sum[StirlingS2[n,k]((x^n)/(n!)),{n,k,∞}]

(∑_(k=m)^∞)(z^k) (StirlingS2[k, m])==((z^m)/((∏_(k=1)^m)(1-k z)))==(((((-z))^(-m)) (z^m))/(Pochhammer[((z-1)/z), m]))

Subtract@@@Table[{
Sum[StirlingS2[k,m]z^k,{k,m,300}],
((z^m)/(Product[1-k z,{k,m}]))
},{m,10}]/.z->.09//Chop//Union//Timing

Sum[StirlingS2[k,m]z^k,{k,m,∞}]//FullSimplify

((z^m)/(Product[1-k z,{k,m}]))

FullSimplify[%,z∈Reals]

(∑_n^∞)(x^n) (StirlingS2[n, k])==((((-1))^k)/(Pochhammer[((x-1)/x), k]))

Product[1-n x,{n,k}]

FunctionExpand[((x^k)/((((-x))^k) Pochhammer[((-1+x)/x),k])),k∈Integers&&k>0]

FullSimplify[((x^k)/((((-x))^k) Pochhammer[((-1+x)/x),k])),k∈Integers&&k>0]

Sum[StirlingS2[n,k](x^n),{n,∞}]

Simplify[Subtract@@@Table[{Sum[StirlingS2[n,k](x^n),{n,∞}],((((-1))^k)/(Pochhammer[((-1+x)/x),k]))},{k,10}]]

Table[StirlingS2[k,5],{k,10}]

(∑_(n=k)^∞)(x^n) (StirlingS2[n, k])==((((-1))^k)/(Pochhammer[((x-1)/x), k]))

Sum[StirlingS2[n,k](x^n),{n,k,∞}]

FullSimplify[%,k∈Integers&&k>=0]

(∑_k^n)(k-1)! (z^k) (StirlingS2[n, k])==(((-1))^n) (PolyLog[1-n, 1+(1/z)])

Sum[StirlingS2[n,k](k-1)!(z^k),{k,n}]

(∑_k^n)(StirlingS2[n, k])==(BellB[n])

V7

Sum[StirlingS2[n,k],{k,0,n}]

Sum[StirlingS2[n,k],{k,n}]

V8

Sum[StirlingS2[n,k],{k,0,n}]

Sum[StirlingS2[n,k],{k,n}]

(∑_(k=0)^n)(x^k) (StirlingS2[n, k])==(BellB2[n, x])

Sum[(x^k)StirlingS2[n,k],{k,0,n}]

(∑_(m=1)^n)(((-1))^m) (m-1)! (StirlingS2[n, m])==0

V6

Sum[(((-1))^m)(m-1)!StirlingS2[n,m],{m,n}]

V7

Sum[(((-1))^m)(m-1)!StirlingS2[n,m],{m,n}]

(∑_(k=m)^n)(m^(n-k)) (Subscript[𝒮, k-1]^((m-1)))

V6

Sum[m^(n-k)StirlingS2[k-1,m-1],{k,m,n}]

V7

Sum[m^(n-k)StirlingS2[k-1,m-1],{k,m,n}]

Table[Sum[m^(n-k)StirlingS2[k-1,m-1],{k,m,n}],{n,4},{m,n}]

(∑_(j=i)^n)(∑_(j=0)^k)(StirlingS2[j, i]) (StirlingS1[k, j]) (StirlingS1[n, k])==(StirlingS1[n, i])

Sum[StirlingS1[n,k]StirlingS1[k,j]StirlingS2[j,i],{k,i,n},{j,0,k}]

Table[Subtract@@{Sum[StirlingS1[n,k]StirlingS1[k,j]StirlingS2[j,i],{k,i,n},{j,0,k}],StirlingS1[n,i]},{n,5},{k,n},{i,k}]

(∑_(k=i)^n)(∑_(j=0)^k)(StirlingS1[j, i]) (StirlingS2[k, j]) (StirlingS2[n, k])==(StirlingS2[n, i])

Sum[StirlingS2[n,k]StirlingS2[k,j]StirlingS1[j,i],{k,i,n},{j,0,k}]

Table[Subtract@@{Sum[StirlingS2[n,k]StirlingS2[k,j]StirlingS1[j,i],{k,i,n},{j,0,k}],
StirlingS2[n,i]},{n,5},{k,n},{i,k}]

(∑_(k=0)^(m-n))(((-1))^k) (Binomial[k+n-1, k-m+n]) (Binomial[2 n-m, -k-m+n]) (StirlingS1[k-m+n, k])==(StirlingS2[n, m])

Sum[(((-1))^k)Binomial[n+k-1,k+n-m]Binomial[2n-m,n-k-m]StirlingS1[k-m+n,k],{k,0,m-n}]

Table[Subtract@@{Sum[(((-1))^k)Binomial[n+k-1,k+n-m]Binomial[2n-m,n-k-m]StirlingS1[k-m+n,k],{k,0,n-m}],
StirlingS2[n,m]},{n,5},{m,n}]

(∑_(k=0)^(m-n))(((-1))^k) (Binomial[k+n-1, k-m+n]) (Binomial[2 n-m, -k-m+n]) (StirlingS2[k-m+n, k])==(StirlingS1[n, m])

Sum[(((-1))^k)Binomial[n+k-1,k+n-m]Binomial[2n-m,n-k-m]StirlingS2[k-m+n,k],{k,0,m-n}]

Table[Subtract@@{Sum[(((-1))^k)Binomial[n+k-1,k+n-m]Binomial[2n-m,n-k-m]StirlingS2[k-m+n,k],{k,0,n-m}],
StirlingS1[n,m]},{n,5},{m,n}]

(StirlingS1[n, n-k])==(Binomial[n-1, k]) (∑_(j=0)^k)(((-1))^j) (Binomial[k+1, j+1])((Binomial[k+j n, k])^(-1)) (StirlingS2[k+j n, j n])

Sum[(((-1))^j)Binomial[k+1,j+1]((Binomial[k+j n,k])^(-1))StirlingS2[k+j n,j n],{j,0,k}]

Refine[(((-1))^k)StirlingCarlitzS1[n-1,k],k∈Integers]

StirlingCarlitzS2[j n,k]

Binomial[n-1,k]Sum[(((-1))^j)Binomial[k+1,j+1]((Binomial[k+j n,k])^(-1))StirlingS2[k+j n,j n],{j,0,k}]

Subtract@@@Flatten[Table[{StirlingS1[n,n-k],Binomial[n-1,k]Sum[(((-1))^j)Binomial[k+1,j+1]((Binomial[k+j n,k])^(-1))StirlingS2[k+j n,j n],{j,0,k}]},{k,10},{n,k,10}],1]//Union

(∑_(l=0)^(max(j,k)+1))(StirlingS1[l, j]) (StirlingS2[k, l])==(KroneckerDeltaSeq[j,k])

Sum[StirlingS1[l,j]StirlingS2[k,l],{l,0,Max[k,j]+1}]

Table[Subtract@@{Sum[StirlingS1[l,j]StirlingS2[k,l],{l,0,Max[k,j]+1}],
KroneckerDelta[j,k]},{j,5},{k,5}]

(∑_(l=0)^(max(k,j)+1))(StirlingS2[l, j]) (StirlingS1[k, l])==(KroneckerDeltaSeq[j,k])

Sum[StirlingS1[k,l]StirlingS2[l,j],{l,0,Max[k,j]+1}]

Table[Subtract@@{Sum[StirlingS1[k,l]StirlingS2[l,j],{l,0,Max[k,j]+1}],
KroneckerDelta[j,k]},{j,5},{k,5}]

(∑_m^n)(((-1))^m) (m-1)! (StirlingS2[n, m])==0

Sum[(((-1))^m)(m-1)!StirlingS2[n,m],{m,n}]

(∑_(k=0)^n)k! (Binomial[m+1, k+1]) (StirlingS2[n, k])==(HarmonicNumber2[m, -n])

Sum[k^n,{k,m}]

Sum[k!Binomial[m+1,k+1]StirlingS2[n,k],{k,0,n}]

Table[Subtract@@@{HarmonicNumber[m,-n],Sum[k!Binomial[m+1,k+1]StirlingS2[n,k],{k,0,n}]},{m,5},{n,5}]

(∑_(k=1)^∞)((((m/(m+1))))^k) (k^n)==(PolyLog[-n, (m/(m+1))])

V6

FullSimplify[Sum[(k^n)(((m/(m+1)))^k),{k,∞}],(m|n)∈Integers]

V7

Sum[(k^n)(((m/(m+1)))^k),{k,∞}]

f[m_,n_]:=PolyLog[-n,(m/(1+m))]

Table[f[m,n],{n,4},{m,n}]

Table[f[1,n],{n,30}]

Union[Mod[%,10]]

(m+1)Sum[k! StirlingS2[n,k] m^k,{k,n}]

Table[(m+1)Sum[k! StirlingS2[n,k] m^k,{k,n}],{n,4},{m,n}]

(∑_(k=0)^∞)(k^n) (k-((Gamma2[k+2, 1])/(Gamma[k+1]))+1)==(∑_(k=0)^n)((StirlingS2[n, k])/(k+2))

K.A. Penson, 10 April 2002

FullSimplify[((k^n)/(k!))((k+1)!-Gamma[k+2,1])]

Table[{
Sum[k^n (1+k-((Gamma[2+k,1])/(Gamma[1+k]))),{k,0,100}]//N,Sum[StirlingS2[n,k]/(k+2),{k,0,n}]},{n,0,3}]

N[%]

Sum[ (1+k-((Gamma[2+k,1])/(Gamma[1+k]))),{k,0,100}]//N

Table[Sum[StirlingS2[n,k]/(k+2),{k,0,n}],{n,0,10}]

V7

Sum[k^n (1+k-((Gamma[2+k,1])/(Gamma[1+k]))),{k,0,∞}]

Sum[StirlingS2[n,k]/(k+2),{k,0,n}]

Recurrence

(StirlingS2[n-1, k-1])+k (StirlingS2[n-1, k])==(StirlingS2[n, k])

FullSimplify[StirlingS2[n-1,k-1]+k StirlingS2[n-1,k],(n|k)∈Integers&&n>0&&k>0]

FullSimplify[Subtract@@{StirlingS2[n-1,k-1]+k StirlingS2[n-1,k],StirlingS2[n,k]},(n|k)∈Integers&&n>0&&k>0]

Table[Subtract@@{StirlingS2[n-1,k-1]+k StirlingS2[n-1,k],StirlingS2[n,k]},{k,10},{n,10}]