Let there be an array of digits $ [0,0,1,1,2,2,\ldots,9,9] $, how many ways to split this array into two subarrays, so that the sum of numbers in the subarrays is different by a multiple of $11$?
This problem has arisen through the solution of some other problem, so I do not cite my attempts at solution. I will be glad to any help.
P.S. If it is difficult to calculate manually, then I will be glad to see an effective algorithm.
I assume that we have eliminated the $0'$s, in line with my comment. Following up астон вілла олоф мэллбэрг's observation, I think a good way to approach it is to figure out all the subarrays that sum to $11$. Then we successively compute the subarrays that sum to $1$, $12$, $23$ etc. We know that there is only subarray that sums to $1$. We can adjoin any of the subarrays that sum to $11$, unless that results in a subarray with more than $2\ 1'$s, to generate the subarray that sum to $12,$ and so on.
I can't see a way to compute the number without constructing the subarrays.