Diameter of the "Masterball-puzzle" permutation groups by a kind of Cartier-Foata enumeration?

Question

There is an wonderful blog post by Jordan S. Ellenberg SHOULD YOU BE SURPRISED BY THE DIAMETER OF THE NXNXN RUBIK’S GROUP?. Which explains how one can come to $N^2log(N)$ estimate of the diameter of the N-dimensional Rubik cube. (See also MO77836). If I understand correctly - one the ideas is incorporation of the huge commutative subsets of the generators (rotations of parallel layers) and employing the Cartier-Foata enumeration for the number of length-k words in a “partially commutative monoid”.

There is another example of the permutation subgroups family denoted "Globe A/B" (for any A,B - positive integers). Which are related to globe ("Masterball"), not cube puzzles. The detailed definition of that family of groups can be found here MSE4848434. The point is that they contain huge family of commuting generators $r_0, r_1, ..., r_{A}$ and the other set of generators of order 2 : $f_0, ..., f_{2B-1}$.

Question: Are there any ways to estimate diameter ("God's number") of such subgroups ? Can the Cartier-Foata counting argument be employed ? Any other information on these groups - welcome to share !

Context: There is ongoing activity on data science platform Kaggle where nearly 1000 participants trying to "crowd-source diameter of these groups", well, more precisely competing to find the shortest path between two given elements (the shorter - the nearer to prize). Interesting, that machine learning methods (e.g.) can actually be applied to these tasks. Imho, interaction between Kaggle-ML and Math communities might be great. GAP computation of the size and some remarks on diameter estimation are here.

PS

Question 2: Can Cartier-Foata counting be generalized to the more general class of relaxed-commutative relations ("Manin's relations") and what can be applications to group theory ?

Cartier-Foata results seems to be based on the fact that for matrices where entries from different rows commute (but may be non-commutative in each row) one can prove many theorems of linear algebra in particular MacMahon type identity: $1/det(1-tA) = \sum Tr_{S^n} A $. Actually there is a more general class of matrices with non-commuting entries - "Manin matrices" where all the theorems of linear algebra holds true despite partial non-commutativity of elements (see also).

Answer 1

Some partial information.

Cartier-Foata - Koszul duality The classical Cartier-Foata counting formula can be seen as particular case of the Koszul duality. Consider algebra generated by several variables which commute, the others are free. Such algebra is Koszul and standard equality $1 = H_A(t)H_A^{dual}(t)$ gives the Cartier-Foata formula.

Generalization precisely including $x_i^2=0$ The "Masterball"("Globe") puzzle contains generators $x_i^2=1$. On the algebra level it is similar to $x_i^2=0$. And that algebra is also Koszul and there is generalization of the Cartier-Foata formula:

Vic Reiner, "Koszul algebras in combinatorics", page 16, https://www-users.cse.umn.edu/~reiner/Talks/kslides.pdf

So seems to be there is a desired generalization of the Cartier-Foata formula for the "Masterball"("Globe") case.

Hopefully it can be used to get some info on the diameter.

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MacMahon identity. The standard MacMahon identity can be seen as a corollary of the standard Koszul duality between polynomials and exterior algebra, where matrix $A$ acts on generators and we calculate the TRACE over the Koszul complex and thus we get: $ 1 = det(1-tA)(\sum t^k Tr_{S^k} A )$.

The point of some modern papers that $A$ may be "Manin matrix", i.e. one can relax commutativity requirment for elements of $A$ and only require that $A$ preserves the relation between the generators.

Manin matrices seems have not been investigated in the case of such generalized Koszul dual pair, to the best of my knowledge.