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1$\begingroup$ I haven't worked through the details, but I expect this would follow fairly immediately from the first (un-numbered) theorem on Page 379 of this paper: the sum described in that theorem also has $(n+1)^{n-1}$ terms, and there is a similar distinction between terms like $-P_{ij}$ and $P_{i1}+\cdots+P_{in}$ there. $\endgroup$Nathaniel Johnston– Nathaniel Johnston2024-11-08 16:42:31 +00:00Commented Nov 8, 2024 at 16:42
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$\begingroup$ @NathanielJohnston Thanks, the suggestion was spot-on. I have filled in the blanks in a CW answer, but if you wish to write your own answer I will be happy to accept it. $\endgroup$Federico Poloni– Federico Poloni2024-11-09 12:51:55 +00:00Commented Nov 9, 2024 at 12:51
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$\begingroup$ Nope, I’m happy to not work through the details myself. Glad it worked out! $\endgroup$Nathaniel Johnston– Nathaniel Johnston2024-11-09 13:48:03 +00:00Commented Nov 9, 2024 at 13:48
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