Probability of getting A to K on single scan of shuffled deck Let us say we have a regular 52-card well-shuffled deck.
We scan through the deck (first to last) till we find an Ace. Then we continue (from that Ace) till we find a 2. Then we scan (from the 2) till we find a 3, and so on. We stop when we find a King. (Suits don't matter.)
What is the probability that we can complete this process within one scan of the deck?
It seems an extremely hard question, and I haven't made any progress. I don't know if this has been answered elsewhere, if so, a link is enough.

Here's a hands-on intuitive approach to the problem.  Suppose we have an alphabet $\mathcal{A}=\{a_1,\ldots,a_k\}$.  Given nonnegative integers $n_1,\ldots,n_k$ and $m_1,\ldots, m_k$, let
$$\left[\begin{array}{c}n_1,\ldots,n_k\\m_1,\ldots,m_k\end{array}\right]$$
 denote the number of words that can be formed from $n_i$ copies of $a_i$, such that the word contains, as a (possibly nonconsecutive) subsequence, a pattern containing exactly $m_i$ copies of the letter $a_i$.  (The count is independent of which pattern is chosen.)  Your problem is to compute
$$\left[\begin{array}{c}4,\ldots,4\\ 1,\ldots,1\end{array}\right]$$
where there are 13 terms on both the top and bottom.
Some special cases are easy: if any $n_i$ or $m_i-n_i$ is negative, the count is zero, and if $k=0$ the count is $1$ (for the empty word).  If $n_i=m_i$ for all $i$, then the count is $1$, since the pattern accounts for the entire word.  Nontrivial values can be computed via two recursive rules:
The Zero Rule:  If $m_i=0$ for some $i$, then
$$\left[\begin{array}{c}n_1,\ldots,n_k\\m_1,\ldots,m_k\end{array}\right]=
  \binom{\sum n_j}{n_i}\left[\begin{array}{c}n_1,\ldots,\hat{n_i},\ldots,n_k\\m_1,\ldots,\hat{m_i},\ldots,m_k\end{array}\right]$$
where the hats indicate that $n_i$ and $m_i$ have been omitted.
To see this, note that if $m_i=0$, the pattern doesn't actually involve the letter $a_i$, so the $a_i$'s can be placed arbitrarily; the binomial coefficient counts the ways to do that.  As a special case, note that when $m_i=0$ for all $i$, the Zero Rule tells us that the count is a multinomial coefficient:
$$\left[\begin{array}{c}n_1,\ldots,n_k\\ 0,\ldots,0\end{array}\right]=\binom{\sum_i n_i}{n_1,\ldots,n_k},$$
as expected since the pattern is empty, so every permutation of the letters matches.
The Peeling Rule: For any $j$, $1\le j \le k$, we can "peel off" the first letter of the word to get:
$$\left[\begin{array}{c}n_1,\ldots,n_k\\m_1,\ldots,m_k\end{array}\right]=
\sum_{i=1}^k \left[\begin{array}{c}n_1,\ldots,n_{i-1},n_i-1,n_{i+1},\ldots,n_k\\m_1,\ldots,m_{i-1},m_i-\delta_{ij},m_{i+1},\ldots, m_k\end{array}\right]$$
where as usual $\delta_{ij}$ is $1$ iff $i=j$.  (To see this, we can assume the pattern starts with $a_j$.  Consider all possibilities $a_i$ for the first letter in the word; it starts a pattern match iff $i=j$.)
To see that these rules suffice, note that peeling produces a sum of terms with strictly smaller $\sum n_i$. 
Finally, note that the count is unchanged if the same permutation is applied to the $n_i$'s and the $m_i$'s.  While this doesn't make the expression simpler, it is useful when using dynamic programming to minimize the number of subexpressions that need to be evaluated.  A straightforward Mathematica implementation verifies the excellent answer provided by @nczkxv.
> S[Array[4 &, 13], Array[1 &, 13]] // Timing
> {1.171875, 50972203946555791528902451677555189167087762981}

