In quantum field theory (specifically $\phi^4$ theory), $W$ is the sum of all connected Feynman diagrams and the effective action $\Gamma$ is the sum of all 1PI Feynman diagrams. They are related by a Legendre transform. Then $W^{(n)}$ and $\Gamma^{(n)}$ represent $n$-point connected correlation functions and $n$-point 1PI functions obtained from $W$ and $\Gamma$ by taking functional derivatives. 

How can one interpret these correlation functions and represent them using diagrams? Is $W^{(n)}$ the sum of all connected diagrams with $n$ external legs? Likewise, is $\Gamma^{(n)}$ the sum of all 1PI diagrams with $n$ external legs? 

There also exists an object $\Sigma$, which is the sum of all 1PI diagrams with 2 external legs (Peskin & Schroeder page 219). Does this mean that $\Sigma = \Gamma^{(2)}$? 

I often see $\Sigma$ written with an argument, for example $\Sigma(p)$ or $\Sigma(p^2)$, is $p$ the momentum associated with the two external legs?

There already exists a similar question (https://physics.stackexchange.com/questions/677065/what-is-diagrammatically-the-2-vertex-gamma) but I do not fully understand the accepted answer, specifically why the diagram they show represents $\Gamma^{(n)}$.