A q-analogue is a sophisticated method to enumerate an object by keeping track of one or more of its mathematical properties. After setting q=1, one returns to the naive enumeration.

After reviewing some classical q-analogues, we will discuss the new idea of a negative q-analogue. This will include recent work of Fu, Reiner, Stanton and Thiem on the negative q-binomial coefficient, and new work on negative q-Stirling numbers of the first and second kind. For the negative q-Stirling numbers we will show Stembridge's q=-1 phenomenon holds both enumeratively and topologically.

This is joint work with Yue Cai.