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  "body": "\n## Definition\n\nLet $x_1, x_2, \\ldots$ be countably many commuting variables, and consider [[formal power series|033WGN00BF]] in them with integer coefficients.\nA *symmetric function* is such a series $f$ with these properties:\n\n1. $f$ has bounded degree: for some integer $d$, every monomial of $f$ has total degree at most $d$;\n2. $f$ is invariant under every permutation of the variables.\n   Thus $x_{i_1}^{a_1} \\cdots x_{i_r}^{a_r}$ and $x_{j_1}^{a_1} \\cdots x_{j_r}^{a_r}$ have equal coefficients when the $i_s$ are distinct and the $j_s$ are distinct.\n\nThe symmetric functions form a ring, written $\\Lambda$.\n\n## Remarks\n\nThe ring $\\Lambda$ is graded, $\\Lambda = \\bigoplus_{k \\ge 0} \\Lambda^k$, where $\\Lambda^k$ consists of the homogeneous symmetric functions of degree $k$.\nFor a partition $\\lambda = (\\lambda_1, \\ldots, \\lambda_r)$, the monomial symmetric function $m_\\lambda$ is the sum of all distinct monomials $x_{i_1}^{\\lambda_1} \\cdots x_{i_r}^{\\lambda_r}$ with distinct indices.\nThe $m_\\lambda$ with $\\lambda$ a partition of $k$ form a $\\Z$-basis of $\\Lambda^k$.\nSetting $x_{n+1} = x_{n+2} = \\cdots = 0$ maps $\\Lambda$ onto the ring of [[symmetric polynomials|0VHSHB9B6J]] in $x_1, \\ldots, x_n$.\nEquivalently, $\\Lambda^k$ is the inverse limit of the degree-$k$ symmetric polynomials in $n$ variables, as $n \\to \\infty$.\n\nExample: the power sum $p_2 = x_1^2 + x_2^2 + x_3^2 + \\cdots$ is a symmetric function of degree $2$.\nBoth clauses hold: all monomials have degree $2$, and a permutation of the variables only permutes the terms.\n\nNon-example: the product $\\prod_{i \\ge 1} (1 + x_i) = 1 + e_1 + e_2 + \\cdots$ is invariant under all permutations of the variables, but it fails clause 1.\nIt contains the monomial $x_1 x_2 \\cdots x_k$ of degree $k$ for every $k$.\n\nThe elementary symmetric functions $e_k = \\sum_{i_1 \u003c \\cdots \u003c i_k} x_{i_1} \\cdots x_{i_k}$ are algebraically independent, and $\\Lambda = \\Z[e_1, e_2, \\ldots]$.\nThe same holds for the complete homogeneous symmetric functions $h_k$, the sum of all monomials of degree $k$.\n",
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