Lexicographically k-th small string

Question

The origin problem is here. Now it is deleted.

Suppose I have 3 'available' copies of a, 2 of b, 3 of c, and 4 of d.

1. I want to know the number of different strings with length $l$ using these characters (but no more than the available copies).
2. Is there an efficient (i.e. $o(k)$) algorithm for computing the $k$-th smallest string of length $l$ in lexicographic order?

For example, The first string of length 1 is a, the second string of length 3 is aab, the 5th string of length 5 is aaacc (following aaabb, aaabc, aaabd, and aaacb), etc.

What kind of algorithm or math can I use to calculate?

Answer 1

You are asking two questions. The first is an enumeration question, and the second is about generation or encoding/decoding. The enumeration question is a standard combinatorial exercise, which can be solved using exponential generating functions (and algorithmically, using dynamic programming). For example, the number of strings of length $\ell$ in your example is $\ell!$ times the coefficient of $x^\ell$ in the exponential generating function $$ \left(1 + x + \frac{x^2}{2!}\right) \left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}\right)^2 \left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}\right). $$

For the generation problem, the idea is as follows. Suppose that we want the $k$th lexciographically smallest string of length $\ell$. We will uncover the letters one by one. To find the first letter, we count how many strings of length $\ell$ start with each letter—an instance of the enumeration problem mentioned above—and use this information to determine the first letter. We find the subsequent letters in a very similar way.