In particular: 1) give authors the ability to use Arabic numerals for parts and 2) allow authors to choose to automatically apply part/chapter numbering, restarting chapter numbers in each part; i.e. 1.1, 1.2, 1.3, 1.4; 2.1, 2.2, 2.3; 3.1, 3.2, 3.3; etc.
For implementation, I'm envisioning the creation of three separate global theme options:
Part numbering options: arabic [1,2,3,4,5, etc.], roman [I, II, III, IV, etc.], none.
Chapter numbering options: arabic or roman. [Chapters can still be designated as numberless within the chapter editor, which keeps them separate from the counter].
If Part numbering options are set to a value other than none, a third is available "Chapter numbering rules" option: continuous [default value, corresponds to the way books are now], restart with each part [they would use the arabic/roman rules for parts/chapters and would look like: 1.1, 1.2, 1.3, 2.1, 2.2, 2.3, etc. OR I.i, I.ii, I.iii, I.iv, II.i, II.ii, II.iii, II.iv, etc. OR I.1, I.2, I.3, II.1, II.2, etc. OR 1.i, 1.ii, 1.iii, 2.i, 2.ii, etc.]
In particular: 1) give authors the ability to use Arabic numerals for parts and 2) allow authors to choose to automatically apply part/chapter numbering, restarting chapter numbers in each part; i.e. 1.1, 1.2, 1.3, 1.4; 2.1, 2.2, 2.3; 3.1, 3.2, 3.3; etc.