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How could we reset enumeration after each part? I.e. how can we have part I with chapters 1,2,3 etc, and then part II with chapter 1,2,3 ?

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    Usually chapters are numbered sequentially ignoring parts; resetting the numbers would make cross-references to the chapters quite clumsy: “in chapter 3 of part 2 we discussed…” – egreg Jul 11 '13 at 11:04
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\makeatletter
\@addtoreset{chapter}{part}
\makeatletter

will do what you ask, although it is perhaps more normal to also do

\renewcommand{\thechapter{\thepart.\arabic{chapter}}

so the chapters are prefixed with the part number, unless the parts are sufficient;y distinct that you never need to cross reference between them.

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