Suppose that we have some finite number of $k$-tuples then we define the lcm of two of these tuples to be the tuple of lcms of the co-ordinates. E.g. $[(9, 10), (5, 18)] = ([9, 5], [10, 18]) = (45, 90)$. Suppose further that these tuples are all tuples which multiply to the same squarefree number $d$ (in the above case this would be $90$ which isn't squarefree, but just run with it).

Fix some $k$-tuple $\textbf{n} = (n_1, n_2, \ldots, n_k)$ such that $n_i \mid d$ for all $i$. How many solutions are there, for each $r$, to

\begin{equation} \operatorname{lcm}\limits_{1 \leq i \leq r}\textbf{m}_i = \textbf{n} \end{equation}

where the $\textbf{m}_i$ are distinct $k$-tuples whose components multiply to give $d$?

It should surely be something which depends only on $\omega(d)$; $r$; and the greatest common divisor of the combinations of the $n_i$. For example, if $\textbf{n} = (6, 6, 6)$ and $d = 6$ then there are no pairs of $3$-tuples satisfying the above conditions such that their lowest common multiple is $(6, 6, 6)$. On the other hand, if $r = 3$ then there are five combinations: \begin{align} (1, 1, 6) (1, 6, 1) (6, 1, 1)\\ (1, 2, 3) (1, 3, 2) (6, 1, 1)\\ (1, 2, 3) (3, 1, 2) (2, 3, 1)\\ (2, 1, 3) (1, 3, 2) (3, 2, 1)\\ (2, 1, 3) (3, 1, 2) (1, 6, 1) \end{align}

Suppose that we have some finite number of $k$-tuples then we define the lcm of two of these tuples to be the tuple of lcms of the co-ordinates. E.g. $[(9, 10), (5, 18)] = ([9, 5], [10, 18]) = (45, 90)$. Suppose further that these tuples are all tuples which multiply to the same squarefree number $d$ (in the above case this would be $90$ which isn't squarefree, but just run with it).

Fix some $k$-tuple $\textbf{n} = (n_1, n_2, \ldots, n_k)$ such that $n_i \mid d$ for all $i$. How many solutions are there, for each $r$, to

\begin{equation} \operatorname{lcm}\limits_{1 \leq i \leq r}\textbf{m}_i = \textbf{n} \end{equation}

where the $\textbf{m}_i$ are distinct $k$-tuples whose components multiply to give $d$?

It should surely be something which depends only on $\omega(d)$; $r$; and the greatest common divisor of the combinations of the $n_i$. For example, if $\textbf{n} = (6, 6, 6)$ and $d = 6$ then there are no pairs of $3$-tuples satisfying the above conditions such that their lowest common multiple is $(6, 6, 6)$. On the other hand, if $r = 3$ then there are five combinations: \begin{align} (1, 1, 6) (1, 6, 1) (6, 1, 1)\\ (1, 2, 3) (1, 3, 2) (6, 1, 1)\\ (1, 2, 3) (3, 1, 2) (2, 3, 1)\\ (2, 1, 3) (1, 3, 2) (3, 2, 1)\\ (2, 1, 3) (3, 1, 2) (1, 6, 1) \end{align}

EDIT: I should have included $(1, 1, 6)(2, 3, 1)(3, 2, 1)$ so there are $6$ different combinations.

Suppose that we have some finite number of $k$-tuples then we define the lcm of two of these tuples to be the tuple of lcms of the co-ordinates. E.g. $[(9, 10), (5, 18)] = ([9, 5], [10, 18]) = (45, 90)$. Suppose further that these tuples are all tuples which multiply to the same squarefree number $d$ (in the above case this would be $90$).

Fix some $k$-tuple $\textbf{n} = (n_1, n_2, \ldots, n_k)$ such that $n_i \mid d$ for all $i$. How many solutions are there, for each $r$, to

\begin{equation} \operatorname{lcm}\limits_{1 \leq i \leq r}\textbf{m}_i = \textbf{n} \end{equation}

where the $\textbf{m}_i$ are distinct $k$-tuples whose components multiply to give $d$?

It should surely be something which depends only on $\omega(d)$; $r$; and the greatest common divisor of the combinations of the $n_i$. For example, if $\textbf{n} = (6, 6, 6)$ and $d = 6$ then there are no pairs of $3$-tuples satisfying the above conditions such that their lowest common multiple is $(6, 6, 6)$. On the other hand, if $r = 3$ then there are five combinations: \begin{align} (1, 1, 6) (1, 6, 1) (6, 1, 1)\\ (1, 2, 3) (1, 3, 2) (6, 1, 1)\\ (1, 2, 3) (3, 1, 2) (2, 3, 1)\\ (2, 1, 3) (1, 3, 2) (3, 2, 1)\\ (2, 1, 3) (3, 1, 2) (1, 6, 1) \end{align}

Suppose that we have some finite number of $k$-tuples then we define the lcm of two of these tuples to be the tuple of lcms of the co-ordinates. E.g. $[(9, 10), (5, 18)] = ([9, 5], [10, 18]) = (45, 90)$. Suppose further that these tuples are all tuples which multiply to the same squarefree number $d$ (in the above case this would be $90$ which isn't squarefree, but just run with it).

Fix some $k$-tuple $\textbf{n} = (n_1, n_2, \ldots, n_k)$ such that $n_i \mid d$ for all $i$. How many solutions are there, for each $r$, to

\begin{equation} \operatorname{lcm}\limits_{1 \leq i \leq r}\textbf{m}_i = \textbf{n} \end{equation}

where the $\textbf{m}_i$ are distinct $k$-tuples whose components multiply to give $d$?

It should surely be something which depends only on $\omega(d)$; $r$; and the greatest common divisor of the combinations of the $n_i$. For example, if $\textbf{n} = (6, 6, 6)$ and $d = 6$ then there are no pairs of $3$-tuples satisfying the above conditions such that their lowest common multiple is $(6, 6, 6)$. On the other hand, if $r = 3$ then there are five combinations: \begin{align} (1, 1, 6) (1, 6, 1) (6, 1, 1)\\ (1, 2, 3) (1, 3, 2) (6, 1, 1)\\ (1, 2, 3) (3, 1, 2) (2, 3, 1)\\ (2, 1, 3) (1, 3, 2) (3, 2, 1)\\ (2, 1, 3) (3, 1, 2) (1, 6, 1) \end{align}