I'm no expert in the field of number theory, if this is the exact field, and I just hope my question is not ill-posed.
I was reasoning about this: is there something like "an index of consecutiveness" describing more events?
Here more details: let's assume I'm in a laboratory, and I am studying the behaviour and the health of two mice, who both suffer from epileptic attacks.
I study them for a month, every day and let's say that a mouse has three consecutive epileptic attacks (for example on the $27$th, $28$th, and $29$th day of the month) and the other one has three attacks too, but only two of them are consecutive (for example on $7$th, $10$th and $11$th day of the month).
Is there a way in which I can express, mathematically, the fact that the three attacks of the first mouse are "more consecutive" than the second one's?
The question seems pretty clear to me, but it it's not, forgive me and ask! Thank you.
Thoughts
The first trivial idea is a vector. A sort of function with $N$ arguments, which are the number of days.
$$\psi(d_1, d_2, d_3, \ldots, d_N)$$
So we can set up a week function or a month function.
Actually my first idea was to take into account not the days, but the difference between the $k$ days and the $k+1$ such that:
$$\psi(\Delta t_1, \Delta t_2, \ldots, \Delta t_N)$$
Where $\Delta t_k = t_{k+1} - t_k$.
Clearly the first one is $\Delta t_1 = t_2 - t_1$ and the last one will be $\Delta t_N = t_{N} - t_{N-1}$.
Now I thought about a sort of booleanity for the $\Delta t_k$, for example $t_k = \{0, 1\}$ according to when we have an attack or not. But doing this would be ambiguous since
$$\Delta t_k = 0$$
could occur for both cases in which $t_k = t_{k+1} = 1$ and $t_k = t_{k+1} = 0$, which is not effective at all.
Hence just shift it into the total $\Delta t_k$ in order to have a simple clear form
$$ \Delta t_k = \begin{cases} & 0 \\ & 1 \end{cases}$$
Where $1$ means attack occurring in two consecutive days, whilst $0$ means that one of the two consecutive days we had one attack and no attack, or no attack and attack, or no attack and no attack.
In this way one could form a vector of ones and zeroes, for example
$$\psi_7 = (0, 0, 1, 1, 1, 0, 1)$$
Yet this writing has to take into account that the elements are not the consecutive days, but their "difference". The above function means that we got three consecutive attacks on the $3-4-5$ days.
This could be a bit confusing though.
Elementary definition may then follow, like the "elementary" norm, which is defined as
$$||\psi_N||^2 = d_1^2 + d_2^2 + \ldots + d_N^2 = \alpha \in [0, N]$$
Yet I am still a bit questioning about the index of consecutiveness. One may define it as $S$, or better as $S_1$, the number of consecutive $1$. But I believe it's just an empirical way to act...