I was told by my professor that α-pyrone is not an aromatic compound, but I count two π-electrons as well as one of the lone pairs on oxygen giving six total which would be one of Hückel's $4n + 2$ numbers. Does the carbonyl contribute to aromaticity?
This answer discusses this situation in connection with the closely related 2-pyridone. The $4n+2$ rule does not apply at all when there are pendant pi-bonding atoms, but in the case of a carbonyl oxygen atom (with its high electronegativity) we can consider a contributing structure where the carbonyl pi bond is polarized towards oxygen: $\ce{\overset{+}{C} -\overset{-}{O}}$. If this polarization leaves a ring satisfying the $4n+2$ rule, this zwitterion becomes a significant contributing structure and the molecule will show aromatic characteristics.
In the case of $2$-pyrone the zwitterion structure leaves six pi electrons in the conjugated ring and thus an aromatic contribution is expected. The proton nmr spectrum[1] shows proton NMR shifts greater than that in ethylene and closer to that in benzene, indicating aromatic character. At the same time Wikipedia reports that 2-pyrone
may participate in a variety of cycloaddition reactions to form bicyclic lactones. For example, it readily undergoes Diels-Alder reactions with alkynes producing, upon loss of carbon dioxide, substituted benzenes.[2]
Thus the compound has mixed aromatic and aliphatic characteristics, a common occurrence with molecules having an aromatic-zwitterion contribution from a pendant carbonyl oxygen.
References
Pirkle,W.H. & Dines,M. (1069). _J.Heterocyl. Chem. 6, 1 (1969).
Woodard BT, Posner G H (1999). "Recent Advances in Diels-Alder Cycloadditions Using 2-Pyrones". Advances in Cycloaddition. 5: 47–83.
