MathDB

Let

a_1, a_2, \ldots , a_{11}

be 11 pairwise distinct positive integer with sum less than 2007. Let S be the sequence of

1,2, \ldots ,2007

. Define an operation to be 22 consecutive applications of the following steps on the sequence

S

: on

i

-th step, choose a number from the sequense

S

at random, say

x

. If

1 \leq i \leq 11

, replace

x

with

x+a_i

; if

12 \leq i \leq 22

, replace

x

with

x-a_{i-11}

. If the result of operation on the sequence

S

is an odd permutation of

\{1, 2, \ldots , 2007\}

, it is an odd operation; if the result of operation on the sequence

S

is an even permutation of

\{1, 2, \ldots , 2007\}

, it is an even operation. Which is larger, the number of odd operation or the number of even permutation? And by how many?
Here

\{x_1, x_2, \ldots , x_{2007}\}

is an even permutation of

\{1, 2, \ldots ,2007\}

if the product

\prod_{i > j} (x_i - x_j)

is positive, and an odd one otherwise.

Problem Statement