Score : 1000 points
Inspired by the tv series Stranger Things, bear Limak is going for a walk between two mirror worlds.
There are two perfect binary trees of height H, each with the standard numeration of vertices from 1 to 2^H-1. The root is 1 and the children of x are 2 \cdot x and 2 \cdot x + 1.
Let L denote the number of leaves in a single tree, L = 2^{H-1}.
You are given a permutation P_1, P_2, \ldots, P_L of numbers 1 through L. It describes L special edges that connect leaves of the two trees. There is a special edge between vertex L+i-1 in the first tree and vertex L+P_i-1 in the second tree.
drawing for the first sample test, permutation P = (2, 3, 1, 4), special edges in green
Let's define product of a cycle as the product of numbers in its vertices. Compute the sum of products of all simple cycles that have exactly two special edges, modulo (10^9+7).
A simple cycle is a cycle of length at least 3, without repeated vertices or edges.
Input is given from Standard Input in the following format (where L = 2^{H-1}).
H P_1 P_2 \cdots P_L
Compute the sum of products of simple cycles that have exactly two special edges. Print the answer modulo (10^9+7).
3 2 3 1 4
121788
The following drawings show two valid cycles (but there are more!) with products 2 \cdot 5 \cdot 6 \cdot 3 \cdot 1 \cdot 2 \cdot 5 \cdot 4 = 7200 and 1 \cdot 3 \cdot 7 \cdot 7 \cdot 3 \cdot 1 \cdot 2 \cdot 5 \cdot 4 \cdot 2 = 35280. The third cycle is invalid because it doesn't have exactly two special edges.
2 1 2
36
The only simple cycle in the graph indeed has two special edges, and the product of vertices is 1 \cdot 3 \cdot 3 \cdot 1 \cdot 2 \cdot 2 = 36.
5 6 14 15 7 12 16 5 4 11 9 3 10 8 2 13 1
10199246