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Let $G$ be a simple graph with vertex set $V$. Stanley defined the $G$-analogs of the symmetric function as follows: For $i \ge 0$, define $$e^G_i = \sum_S \big(\prod_{v \in S}v\big)$$ where the sum runs over independent /stable subsets of size $i$ in $G$.

Since the classical elementary symmetric functions $e_{i}$ form a basis for the ring of symmetric functions $e_{i}$ goes to $e^G_{i}$ defines a homomorphism from ring of symmetric functions to $\mathbb{Z}(V)$. Using this homomorphism, we can define $G$-analogs of other classical symmetric functions also.

I want to learn more about this combinatorial object. Kindly suggest some references. Thank you.

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    $\begingroup$ One good place to look would be the 41 papers that MSN knows reference Stanley's. $\endgroup$ – LSpice Jul 2 at 4:49

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