Easier than you may think!

Computer Science Level pending

Suppose we have a \(n\)-tuple \(d=(0,2,2,\dots,n-2,n-2,n)\) for a certain even integer \(n\).

\(d\) represents the list of degrees of the \(n\) nodes in a undirected graph.

Suppose \(n=2^{2^6}-2,\) how many distinct graphs with \(n\) nodes have a permutation of \(d\) as list of degrees?

Give the answer modulo \(10^9+7\).

Note:

The degree of a node in a graph is the number of edges connected to it. Every pair of vertices is connected by at most one edge. Every node cannot be connected with itself.

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