How is OEIS sequence A120933 'maximal leading nondecreasing subword ' to be understood?

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For n=2 we only have these four binary words:

00 01 10 11

What is the procedure for calculating by hand T(2,1) and T(2,2)?

I'm trying to understand the reasoning behind this sequence as I can't see the logic behind it at n=2 much less for n=>3.

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00 starts with 00 which is non-decreasing so k=2
01 starts with 01 which is non-decreasing so k=2
10 starts with 1, but 10 is not non-decreasing, so k=1
11 starts which 11, so k=2
Three of them have k=2, one has k=1, so T(2,1)=1,T(2,2)=3