I have an $n$-tuple $(s_1, s_2, \dots , s_n)$ where all $s_i \in \mathbb{N}\cup \{0\}$. An activation of the starting tuple may cause each element to increase by $1$, remain constant, or decrease by $1$, although not below $0$. Each tuple that is created in an activation becomes a member of the set of tuples $T$. The initial tuple is activated and creates a successor which becomes a member of $T$, then the successor is activated and so on.
Let there be a special criterion that is satisfied iff each element of a newly created tuple $t_1$ is $\geq$ each element of a previously created tuple $t_2$ in $T$ and at least one element of $t_1$ is truly greater than the element at the same position in $t_2$.
For example, the following image shows the case where the initial tuple is $(3, 0, 0)$ which generated a sequence of tuples. Each element of the tuple $(2, 2, 0)$ is $ \geq $ each element of $(2,1,0)$ and the second entry is larger.
- Question: How many mutually distinct tuples can I create at most without satisfying the criterion? The upper bound does not need to be very accurate.
Obviously, if a newly created tuple $t_1$ is to satisfy the criterion with another tuple $t_2 \in T$, then the sum of elements of $t_1$ must be larger than the sum of elements of $t_2$. I was thinking about the number of possibilities to create tuples of decreasing sum of elements. Since a newly created tuple must have a larger sum of elements than a predecessor to satisfy the element, this would be a strategy to create a set of tuples. However, I am not sure if this is the method to create the largest set of tuples.