Google-sheets – Use arrayformula to count columns in rows

In addition to the solution provided by OnenOnlyWalter, here's another formula:

=COUNTIF(ARRAYFORMULA(YEAR(A:A)),"2013")

The YEAR function simply returns the year in 4 digits. The ARRAYFORMULA will just take on the complete column and the COUNTIF will finish it off by counting the occurrences.

If you're using a header, then you can adjust the above formula like this:

=COUNTIF(ARRAYFORMULA(YEAR(A2:A)),"2013")

In addition to previous posts, I think I would go for a query function. Although not that short , it will eliminate the need 1) to drag the current formula down (newly added rows will 'automatically be added to the output of the query) and 2) to have the unique formula to populate col A on the count sheet.

=query(Incidents!B2:C; "select B, count(C) where B<>''  group by B label B 'Name', count(C) 'Number of incidents' ")

EDIT: Closer to the topic, if a list of names is already given (in col A), a count in arrayformula should be possible by doing:

=ArrayFormula(countif(split(concatenate(if(len(Incidents!B2:B),Incidents!B2:B&char(9),)),char(9),0), filter(A:A, len(A:A))))