Here it is, from ARRL:
Good morning,
The serial number is a sequential number, starting with 1 for your first contact, 2 for your second contact, and so on. In the case of the 10 Meter contest, stations located in the US, Canada, and Mexico send their state or province abbreviation, not a serial number.
In the ARRL International DX contest, as a US station, you'd be sending a signal report and your state abbreviation. It's customary to just give a 59 (or 599 if CW) as a signal report and then your state. The DX stations will send a signal report and their serial number, when then goes in your log for cross checking (we check to see if you receive the number they sent you correctly by comparing the logs.)
I can add that term to the glossary which is located at the end of the contest rules PDF files on the ARRL website.
73,
Paul Bourque, N1SFE
Contest Program Manager