Suppose we have a multiset $M$, which contains $k$ distinct elements. Each element $x_i$ has multiplicity $n_i$ for each $i\in\Bbb{N}$ such that $0\le i<k$. $n$, the number of elements in $M$ including repetition, is defined as $n=\underset{i=0}{\overset{k-1}{\sum}}n_i$.

How would I go about calculating the number of permutations of length $r$ of $M$, where each element $x_i$ is repeated $t_i$ times for $0\le t_i\le n_i$? (The specific values of each $t_i$ may vary for each permutation, so long as they all add up to $r$.) I have found this question answered in several places with the additional constraint that $t_i>0$ (which gives an answer of $\frac{n!}{\underset{i=0}{\overset{k-1}{\prod}}t_i!}$), but never in this general case.

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    $\begingroup$ Are you counting lists of length $r$, where each entry is equal to $x_i$ for some $0\le i<k$, and $x_i$ appears $t_i$ times in the list? If so, the answer should be $r!/(\prod_{i=0}^{k-1}t_i!)$, and this should work even without the assumption $t_i>0$. $\endgroup$ – Mike Earnest Feb 8 at 19:20
  • $\begingroup$ Or are you not counting permutations with fixed $t_i$, but instead counting permutations where the $t_i$ can be anything except $n_i$? In other words, permutations where no element is completely used? $\endgroup$ – Mike Earnest Feb 8 at 19:34
  • $\begingroup$ @Mike $t_i$ is not fixed but may vary for each list (such that all of them sum to r). Also, I meant to say that $t_i\le n_i$, not that $t_i<n_i$. I have updated the question accordingly. $\endgroup$ – Evan Bailey Feb 11 at 5:18

I do not think there is a nice formula, but there is a generating function solution.

Let $E_n(x)=\sum_{j=0}^n\frac{x^j}{j!}$ be the partial exponential series. The number of $r$-permutations is $$ r![x^r]\prod_{i=0}^{k-1} E_{n_i}(x) $$ where $[x^r]f(x)$ is the coefficient of $x^r$ in the polynomial $f(x)$.

On a side note, I disagree with the formula $n!/\prod n_i!$ for the number of $r$-permutations where each object appears at least once (each $t_i>0$). First, this does not involve $r$ at all. Second, in the case where the multiset is $\{A,A,B,B\}$ and $r=2$, the answer should be two, since the valid permutations are $AB$ and $BA$, but your formula gives $4!/(2!\cdot 2!)=6$. Instead, $n!/\prod n_i!$ gives the number of $n$-permutations of a multiset with $n$ elements total (all objects used completely, each $t_i=n_i$).

  • $\begingroup$ Thank you for this answer. It was exactly what I was looking for. Also, I will correct the formula in my original question to use $t_i$ in the denominator instead of $n_i$. $\endgroup$ – Evan Bailey Feb 11 at 19:49
  • $\begingroup$ Also, I would assume (although I am no expert when it comes to generating functions) that in this context, it would be equivalent to define $E_n(x)$ as $\frac{x^{n+1}-1}{x^2-x}$. I am just wondering if it would really be simpler in this context to do so. $\endgroup$ – Evan Bailey Feb 11 at 19:56
  • $\begingroup$ @EvanBailey No, that would not work. It is important you use exponential generating functions, meaning the $i!$ in $\frac{x^i}{i!}$ is important. This ensures that order matters, because you are counting permutations where order matters. You can take a look at the exponential generating function chapter of generatingfunctionology for some more details. $\endgroup$ – Mike Earnest Feb 11 at 20:00
  • $\begingroup$ Thank you. I will be sure to check it out. $\endgroup$ – Evan Bailey Feb 11 at 20:44
  • $\begingroup$ So would the generating function for the number of $r$-combinations be the same as that for the number of $r$-permutations sans the $r!$? $\endgroup$ – Evan Bailey Feb 19 at 20:23

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