1. a1=L 2. b1=L 3. c1=L 4. Td6 5. Lf7 6. Tgf6 7. Lg6 8. Tde6 9. Lf4 10. Le4 11. Lxe5 g4#
The 11 moves are determined, but the order is not, and there are some dependencies between them. The underlying partial ordering is termed the "zigzag" (a < b > c < d > ... etc).
The linear extension of this poset ("partially ordered set"), i.e. the number of total orderings which respect the given partial ordering, is the Euler number, E_n, defined by the generating function:
sec(x) + tan(x) = Sum(n=0...infinity) E_n (x^n/n!)
See the Online Encyclopedia of Integer Sequences, https://oeis.org/A000111.
Here the partial ordering is:
a1=L < Lxe5 > b1=L < Le4 > c1=L < Lf4 > T1d6 < Tde6 > Lf7 < Lg6 > Tgf6
and the number of total ordering is E_11 = 353,792.
The problem is not just about combinatorics - it's also about chess. There are a number of tries buried in the position, particularly for mates with Kg5 or Kh5.