Given a positive integer $h$, define: $$A_h=[2^h,2^{h}-1]\big \{2^h-1+\sum_{i\in A}2^i \Big/ A\subset[0,h-1]\big \}$$ (this is in terms of binary expressions: the set of all numbers having exactly $h$ digits in their binary operations)
Given a tuple $(a_1,\cdots,a_n)$, an operation consists of replacing some $a_j$ by either $2a_j,2a_j+1$ or by $a_k$ with $a_k\neq a_j$.
My question:
What is the maximum number of operations which could transform $(0,0,0)$ to some element $(a,b,c)$ with $a,b,c\in A_h$ such that all integers appearing in the intermediates tuples are less then or equal $2^h-1$ and no tuple appears twice in the process.
Exemple
It's clear that we can transform $(0,0,0)$ into $(2^h,2^h,2^h)$ in $h+3$ operations: $$(0,0,0)\xrightarrow{a:=2a+1} (1,0,0)\xrightarrow{a:=2a}(2,0,0)----\cdots \xrightarrow{a:=2a}(2^h,0,0)\xrightarrow{b:=a}(2^h,2^h,0)\xrightarrow{c:=a}(2^h,2^h,2^h)$$ this process is valid because all intermediate tuples have elements less than or equal to $2^h$ and no tuple appears two times. But I'm interested of the worst case : what would be the very long sequence using the operation?
Result: when we use pairs $(a,b)$ instead of tuples we have the maximum number of operations is $\frac{(h+1)(h+2)}{2}-1$.
A graph theory reformulation of the problem Let : $$ A_h=[2^h,2^{h}-1]\ \ \ \ B_h=[0,2^h-1] \\ V_h=B_h^3$$
Given $x=(a,b,c), x'=(a',b',c')\in V_h^3$ we have $(x,x')\in E_h$ if and only if there exists a rearrangement $a_1,a_2,a_3$ of $a',b',c'$ such that $a_1=a,a_2=b',a_3\in \{2a_3,2a_3+1,a_1,a_2\}$
Question: what would be the maximum length of a path between $(0,0,0)$ and $A_h^3$
Note this is a reformulation of my problem: how to maximize the number of operations in a process