The Turkey Ratio

This problem comes from something we do each Thanksgiving. This past Thanksgiving we had 19 (N) people at dinner. Each person writes something for which they are thankful and puts the slip of paper into a bowl. The slips are mixed up and each person draws a slip. This past holiday, someone got their own slip. My question is given an N (19 for example) what is the probability no one gets their own slip? I give a solution below but have some questions (perhaps you'd like to try to solve it without peeking at the answer I provide). Perhaps you can find a more elegant solution. Here are my questions: Is there a name for this problem? The probability oscillates which surprises me; can you offer a similar problem where the problem oscillates? By the way the probability converges a N increases to what I am calling the Turkey Ratio. Does this value appear elsewhere?

So here are the probabilities for 2 to 25:
2: 0.5
3: 0.66666666666666674
4: 0.625
5: 0.6333333333333333
6: 0.63194444444444442
7: 0.63214285714285712
8: 0.63211805555555556
9: 0.63212081128747788
10: 0.63212053571428573
11: 0.63212056076639411
12: 0.63212055867871841
13: 0.63212055883930884
14: 0.63212055882783802
15: 0.63212055882860274
16: 0.632120558828555
17: 0.63212055882855789
18: 0.63212055882855767
19: 0.63212055882855767
20: 0.63212055882855767
21: 0.63212055882855778
22: 0.63212055882855767
23: 0.63212055882855767
24: 0.63212055882855789
25: 0.63212055882855767

So here is the process to generate the probability given that there are n guests.
1. Create a list of the factorials from n! to 0!. (For n=4, we’d have 24, 6, 2, 1, 1)
2. Save first value in the list (24).
3. Create a list of the difference in successive values of the list. (18, 4, 1, 0)
4. Save the first value in the list (18).
5. Repeat steps 3 to 4 until you have saved n+1 values (24, 18, 14, 11, 9)
6. The answer is 1 – the last number divided by the first number (0.625 = 1 - 9/24)

Your turkey ratio is simply $1 - \frac{1}{e}$ = 0.63212055882... (as N approaches infinity)