Is there any result known about counting the number of (unlabeled) ordered trees which follow a given unordered degree sequence?
Here an ordered tree is understood as a rooted tree in which the order of the subtrees is significant.
Is there any result known about counting the number of (unlabeled) ordered trees which follow a given unordered degree sequence? Here an ordered tree is understood as a rooted tree in which the order of the subtrees is significant. 


Is Theorem 6.4 of http://people.brandeis.edu/~gessel/homepage/papers/enum.pdf what you want? 


See Theorem 4.3 on page 12 here. 

