Elementary Symmetric Polynomial - Examples

Examples

The following lists the n elementary symmetric polynomials for the first four positive values of n. (In every case, e0 = 1 is also one of the polynomials.)

For n = 1:

For n = 2:

\begin{align} e_1(X_1,X_2) &= X_1 + X_2,\\ e_2(X_1,X_2) &= X_1X_2.\,\\ \end{align}

For n = 3:

\begin{align} e_1(X_1,X_2,X_3) &= X_1 + X_2 + X_3,\\ e_2(X_1,X_2,X_3) &= X_1X_2 + X_1X_3 + X_2X_3,\\ e_3(X_1,X_2,X_3) &= X_1X_2X_3.\,\\ \end{align}

For n = 4:

\begin{align} e_1(X_1,X_2,X_3,X_4) &= X_1 + X_2 + X_3 + X_4,\\ e_2(X_1,X_2,X_3,X_4) &= X_1X_2 + X_1X_3 + X_1X_4 + X_2X_3 + X_2X_4 + X_3X_4,\\ e_3(X_1,X_2,X_3,X_4) &= X_1X_2X_3 + X_1X_2X_4 + X_1X_3X_4 + X_2X_3X_4,\\ e_4(X_1,X_2,X_3,X_4) &= X_1X_2X_3X_4.\,\\ \end{align}

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