My cry for help on locating a use of !n (which I consider to be the most usual symbol currently in use for the number of derangements of n things) was answered in the most usual of places. My great research pal, Dave Renfro, who has read more journals than any three people I know, sent me a link to:
which comes from the questions section of the MAA in 1958. It was obvioulsy not an immediate success, as I also found the use of Chrystal's inverted exclamation mark in "A Note on Derangements", by M. T. L. Bizley © 1967 The Mathematical Association. A UK based association. If any of my British readers know what symbol is currently in use in the UK I would appreciate a note.
Dave also sent me a copy of the (original?) use by Whitworth of his symbol. It clearly indicates that he was modeling it on the Jarrett symbol for factorial...
Even though he was writing only about 40 years after the symbol was created, he was mistaken in assuming that the name came after the symbol. The word factorial was the creation of Louis François Antoine Arbogast Who died in 1803. The symbol now commonly used for factorial seems to have been created by Christian Kramp in 1808, and Jarrett, who created the symbol that Whitworth thought might have been the forerunner of the name, was born two years after Arbogast died, making him three years old when Kramp first used !n. Jarrett invented his symbol in 1827. It occurs in a paper "On Algebraic Notation" that was printed in 1830 in the Transactions of the Cambridge Philosophical Society.
I should add that it is often the case in modern usage, that writers choose to ignore all these symbols for a function approach, with the most common being a D(r), perhaps for Derangements. If you have a book from any period that addresses the topic, I would be grateful for a short comment telling me what symbol (if any) they used, and the year, title and author... (and if your symbol is not shown here, or your date of !n preceedes the 1958 one above, and electronic image would earn my deep gratitude and a mention in some future post).