Symmetric Polynomial

Author

Eric W. Weisstein
April 14, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/SymmetricPolynomial.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<DiscreteMath`;
<<Algebra`SymmetricPolynomials`

Π[n_,k_:5]:=Plus@@Times@@@(Map[a,KSubsets[Range[k],n],{2}])
s[n_,k_:5]:=Module[{i},Sum[a[i]^n,{i,k}]]

Table

Table[SymmetricPolynomial[Array[x,n],k],{n,3},{k,n}]//ColumnForm

Generating Function

Collect[Expand[Product[x+(Subscript[x, i]),{i,4}]],x]

Determinant Identity

KSubsets[Range[5],2]

Take[Expand[Table[n,{n,0,20}]CoefficientList[Series[Log[1+(Subscript[h, 1])x+(Subscript[h, 2])x^2+(Subscript[h, 3])x^3+(Subscript[h, 4])x^4+(Subscript[h, 5])x^5+(Subscript[h, 6])x^6+(Subscript[h, 7])x^7],{x,0,20}],x]],6]//ColumnForm

smat[m_]:=Table[If[j==1,i,1](Subscript[h, i-j+1]),{i,m},{j,m}]/.(Subscript[h, 0]):>1/.(Subscript[h, n_?Negative]):>0
s[m_]:=(-1)^(m-1)Det[smat[m]]

smat[4]//MatrixForm

Table[s[i],{i,5}]//ColumnForm

Table[Expand[(-1)^(k-1)s[k]/.(Subscript[h, n_]):>Π[n]],{k,5}]//ColumnForm

Power Sums

Π[1]

Expand[Π[1]^2-2Π[2]]

Expand[Π[1]^3-3Π[1]Π[2]+3Π[3]]

Expand[Π[1]^4-4Π[1]^2Π[2]+2Π[2]^2+4Π[1]Π[3]-4Π[4]]

Newton-Girard Formulas

1≤k≤n

NewtonGirard[n_,i_]:=Plus@@Append[Table[(-1)^k s[n-k,i]Π[k,i],{k,0,n-1}],(-1)^n n Π[n,i]] /; 1≤ i≤n

Table[NewtonGirard[n,1],{n,10}]//Expand

Table[NewtonGirard[n,5],{n,10}]//Expand

Table[NewtonGirard[n,10],{n,10}]//Expand

NewtonGirardSym[n_]:=Plus@@Append[Table[(-1)^k S[n-k]P[k],{k,0,n-1}],(-1)^n n P[n]]

(Table[NewtonGirardSym[n],{n,4}]/.P[0]->1)//ColumnForm

k>n

NewtonGirard[n_,k_]:=Plus@@Prepend[Table[(-1)^(i) s[k-i,k]Π[i,k],{i,n}],s[n,k]] /; k>n

NewtonGirard[n_,k_]:=Prepend[Table[(-1)^(i) s1[k-i,k]Π1[i,k],{i,n}],s1[n,k]] /; k>n

NewtonGirard[3,4]

s[0,2]