Permutations of the positive integers

[Kimberling, Clark]

Dear Seqfans -
 
Suppose p is a permutation of the positive integers, N.  If we subtract
1 from every term and then delete 0, what's left is another perm.
Iterate, and we get many perms.
 
Now, let t(k) be the position of 1 in the kth iterate.
 
Example:  p = (1,3,2,5,7,4,9,11,6,13,15,8,...) = A006369 (related to
3X+1 problem)
This choice of p yields trace sequence t = (1,2,1,3,1,4,1,5,1,6,1,7,...)
= A057979 essentially.
 
A sequence t of positive integers is the trace of a perm if and only if
t has infinitely many 1's.  Let T be the set of all such t.
 
Suppose t_p and t_q are traces of perms p and q.  Can someone find a
decent formula for t_r, where r is the composite perm p-of-q?
 
I'd like to see such a formula - it would define a group operation on
the set T. 
 
Clark Kimberling       
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