2013 HMMT November General Round Problem 10

How many functions f : \{1,2,\ldots,2013\} \to \{1,2,\ldots,2013\} satisfy f(j) < f(i)+j-i for all integers i,j such that 1 \le i < j \le 2013?

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2013 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.