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In 2010, Dong, Hue, et all. find the formula

$$S_{Diagram} = g 2^\beta 2^d \prod_n (n!)^{\alpha_n}$$

where we have an interpretation about each of these numbers. In this equation the symmetry factor $S$ is a function $S = S(g,\beta,d,\alpha_n)$ where

  1. $\beta$ is the number of lines that connect a vertex to itself
  2. $d$ is the number of double-bubbles in the diagram
  3. $\alpha_n$ is the number of vertex pairs with $n$ same contractions

My question is, how can I understand $g$ in this formula.

The text uses a lot of examples but to me it is not clear in the examples and in the text what the authors are considering in this $g$.

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Perhaps the notation used is a little misleading but as stated in the article on p.4,

"$\dots g$ is the number of permutations of vertices that leave the diagram unchanged with fixed external lines."

This is to say that $g$ is the number of viable permutations one can make that result in a topologically indistinct diagram. In particular, vertices connected directly to external lines are not subject to interchange because those lines originate from labelled and distinguishable source points, which define the distinct entries in your $n$-point correlators, each at some coordinate $x_i$.

The trivial case of $g=1$ corresponds to the identity permutation, i.e there is no non-trivial permutation one can perform on the diagram that will result in a topologically indistinct diagram. This is all exemplified on p.8-9 of the text, where they give diagrams pertaining to the zero and two point correlator with $g=1$ and $g=2$.

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