Elven mystical numbers

Before the problem itself, it would be a good idea to explain what an "Elven mystical number" is (EMN in the further text). Each EMN has the following properties:

  • (1) Number of digits is odd

  • (2) The middle digit is the peak and it is the greatest digit in number, also it appears only once

  • (3) Going from the peak to one of the ends, numbers decrease or stay the same

  • (4) These numbers are also palindromes

Now, when the EMN is not so mystical, the question is, how many 11-digit EMNs exist?

Details and assumptions:

  • Middle digit is at the same "distance" from left and right end, so in 5-digit number, middle digit is the \( 3^{rd} \) one.

  • Example of EMNs: 12321, 7778777, 111575111, ...

  • Example of non-EMNs: 11111 (rule (2)), 123321 (rule (1) and (2)), 1325231 (rule (3)), 1237521 (rule (4))

  • If we have 11-digit number, whose first digit is null, then we actually don't have an 11-digit number.

If there is any question related to "Elven Mystical Number", it can be asked here.


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