The Subfactorial and Some Historical Notes

Subfactorial the name subractorial was created by W. A Whitworth in The Messenger of Mathematics in May of 1877.  The symbol for the subfactorial is !n, a simple reversal of the use of the exclamation for n-factorial. This was not the symbol used by Whitworth, as at this time many people preferred what is called the Jarret symbol for the factorial. Whitworth added an extra line in the L to make the subfactorial. This symbol for the factorial persisted into the 1950's.  (Notes on the History of the factorial and its symbols.)

.The subractorial, or derangement is about counting the number of ways to take objects which have some order, and arranging them so that none is in its right ordered place.  The numbers 1, 2, 3 can be arranged for example, as 2,3,1, or 3,1,2.  The problem was first considered by Pierre Raymond de Montmort in 1708, and first solved by him in 1713. 

Cajori mentioned the use by George Chrystal (1851-1911) of a subfactorial symbol using N with an upside down exclamation point, but does not mention at all the !n that is the common present symbol, leading me to believe it was created after 1929.

Crystal's books into the fifties continued to use the inverted exclamation symbol and the National Academy of Sciences used the symbol in 1967.
The earliest used of the !n symbol I have ever found is from 1958, In the MAA questions section:

This was obviously not an instant hit, as I received several comments like the following after a post in 2009.
"  I have several books on my shelf, none of which use !n notation.
D(n)
- Matoušek and Nešetřil, 1998
- Niven, 1965. I teach from this book.

D_n
- Chen and Koh, 1992. Interestingly, they use the notation D(n,r,k) to denote the number of r-permutations of N_n with k fixed points, and (good for them) cite Hanson, Seyffarth and Weston 1982 as the originators of this notation.
- Martin, 2001

- d_m
Goulden and Jackson, 1982.-----------------------------------------------------------------------------------------
The Niven book is his well known Mathematics of Choice, and he uses the symbol D(n) . In 1997 Robert Dickau used\$D_n\$ for derangements, another common name for subfactorials.    John Baez used !n  in 2003 without indicating that it was an uncommon symbol.
The formula for subfactorial, also called derangements of a set, is given by \$!n = n!( 1- \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!}..... \frac{1}{n!} )\$,  The quick approximation is !n = n!/e.