2874. Generic Cow Protests

Farmer John's N (1 <= N <= 100,000) cows are lined up in a row and

numbered 1..N. The cows are conducting another one of their strange

protests, so each cow i is holding up a sign with an integer A_i

(-10,000 <= A_i <= 10,000).

FJ knows the mob of cows will behave if they are properly grouped

and thus would like to arrange the cows into one or more contiguous

groups so that every cow is in exactly one group and that every

group has a nonnegative sum.

Help him count the number of ways he can do this, modulo 1,000,000,009.

By way of example, if N = 4 and the cows' signs are 2, 3, -3, and

1, then the following are the only four valid ways of arranging the

cows:

(2 3 -3 1)

(2 3 -3) (1)

(2) (3 -3 1)

(2) (3 -3) (1)

Note that this example demonstrates the rule for counting different

orders of the arrangements.