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February 2022 edit
I suspect that the variables r and s mentioned in the third sentence of the definition refer to the number of indices in the set I and the number of indices in the set J respectively. I am going to make a minor edit and add a phrase indicating this. Rob.Corless (talk) 02:26, 24 February 2022 (UTC)
Moving on to the next paragraph: I came to this page to find out what compound matrices were. I know a fair amount of linear algebra, but not this. I find this paragraph confusing, mostly because I do not know what is meant by "lexicographic order" of the subsets I and J. A link to the Wikipedia article on lexicographic order might help, except that that article states that there are several conventions in use. I suspect that the "combinatorics" variant is the one intended here. I may need to make more substantial edits than above to clarify this paragraph (once I learn what compound matrices are, which is what I came here for; but I shall go read the Handbook of Linear Algebra instead). There is a somewhat more detailed description in Section 4.2 there.
The fact that the variable r has been re-used also adds a small amount to the confusion. I have not yet made edits here and would be happy for another to do it, but I shall try to come back here with a clearer description. Rob.Corless (talk) 02:47, 24 February 2022 (UTC)
I will add a link to the "lexicographic order" page; the definition is clear enough. The combinatorial variant (shortlex) is not needed after all. I might also add a reference to Gantmacher if it isn't there already and to the Handbook of Linear Algebra if it isn't there already. Rob.Corless (talk) 00:44, 28 February 2022 (UTC)