By 'naturally weak squares' they mean Pawns that are either undefended or defended only by the King. For example, SP679 QRBKNNRB has two naturally weak squares: the Pawns on the e-file and the f-file. The e-Pawn is protected only by the King, while the f-Pawn is unprotected.
I covered Undefended Pawns in Chess960 Start Positions some time ago, but never considered Pawns that are defended only by the King. To investigate this further I dragged my chess960 database of SPs out of storage and developed the necessary queries.
It turns out that there are 472 SPs with at least one Pawn defended only by the King. The first in the list of SPs is SP008 QBNNBRKR, where you can easily see that the g-Pawn is defended only by the King, which is sandwiched between its Rooks and which has no Knight on the e-file to cover the g-Pawn. The traditional start position SP518 RNBQKBNR, with its notoriously weak f-Pawn belongs to this group, as does its twin SP534 RNBKQBNR.
Although I expected to find a few SPs with two Pawns defended only by the King, I was surprised to discover that there are exactly 48 of them. The first is SP097 BQNBRNKR, where the King is the only piece protecting the f- & g-Pawns.
How are the 520 Pawns (472 SPs with at least one Pawn plus 48 with two) distributed over the files? Like this...
...The first entry means that, across all 960 SPs, there are a total of 78 b-Pawns protected only by the King. The rules of chess960 preclude any a- or h-Pawns protected only by the King, but it's not immediately obvious why there should be more center Pawns on the list or why the counts increase when moving from the b/g-files to the center files.
The next step in this analysis will be to consider undefended Pawns together with Pawns defended only by the King. I'll do that in my next post.