Derangements with repetition

Question

Hi all,

given (a1,...,an) formed by distinct letters, it's a well known problem to count the number of permutations with no fixed element.

I've been trying to solve a generalization of this problem, when we allow repetition of the letters.

I was able only to partially solve the problem when we have only repetition of a single letter.

If we have n letters and only one of them is repeated p times, then the number O(n,p) of permutations with no fixed element is given by the following recursive relation:

$O(n,0)=O(n,1)=\mbox{Derangement}(n)$

$O(p+1,p)=\dots=O(2p-1,p)=0, O(2p,p)=p!$

$O(n,p)={n-p\choose p} p!\sum_{k=0}^p{p \choose k}O(n-p-k,p-k)$

Does anybody know where this problem have been studied before? Does anybody know a general solution for this problem?

Thanks in advance.

Answer 1

The formula based on Inclusion-Exclusion for the usual number $D(n)$ of derangements of $n$ objects can be generalized. The result is the following.

Fix $k\geq 1$. Let $\mathbb{N}=\lbrace 0,1,2,\dots\rbrace$. For $\alpha=(\alpha_1,\dots,\alpha_k)\in\mathbb{N}^k$, let $D(\alpha)$ be the number of fixed-point free permutations of the multiset with $\alpha_1$ 1's, $\alpha_2$ 2's, etc. Write $x^\alpha = x_1^{\alpha_1}\cdots x_k^{\alpha_k}$. Then $$ \sum_{\alpha\in\mathbb{N}^k} D(\alpha)x^\alpha = \frac{1}{(1+x_1)\cdots (1+x_k)\left(1-\frac{x_1}{1+x_1}-\cdots - \frac{x_k}{1+x_k}\right)} $$ $$ = \frac{1}{1-\sum_S (|S|-1)\prod_{i\in S}x_i}, $$ where $S$ ranges over all nonempty subsets of $\lbrace 1,2,\dots,n\rbrace$.

This result appears as Exercise 4.5.5 in Goulden and Jackson, Combinatorial Enumeration. It can be used to obtain a lot of information about $D(\alpha)$.

Answer 2

For the particular case you asked about, in which we're computing fixed-point free permutations of the word 0...012...n (where 0 appears p times):

We choose the positions for the 0s in $\binom{n}{p}$ ways. Given this choice, the permutations in question are equivalent to permutations of [n] in which the first n - p values are not fixed points. The OEIS (specifically, sequence A047920) and/or a simple argument by inclusion-exclusion gives $\sum_{ j\geq 0} (-1)^j \binom{n - p}{j}\cdot (n-j)!$ for this value; the OEIS also gives $\sum_{j = 0}^{p} D_{n-j}\cdot \binom{p}{j}$ (where $D_m$ is a derangement number).

Another case that might be of interest is the case in which each letter appears the same number of times. This is discussed in http://academic.csuohio.edu/bmargolius/homepage/dinner/dinner/cardentry.htm and has several associated OEIS entries (e.g., A000459, A059072 and A059073).