  We study superconformal indices of 4d N=2 class S theories with certain
irregular punctures called type $I_{k, N}$. This class of theories include
generalized Argyres-Douglas theories of type $(A_{k-1}, A_{N-1})$ and more. We
conjecture the superconformal indices in certain simplified limits based on the
TQFT structure of the class S theories by writing an expression for the wave
function corresponding to the puncture $I_{k, N}$. We write the Schur limit of
the wave function when $k$ and $N$ are coprime. When $k=2$, we also conjecture
a closed-form expression for the Hall-Littlewood index and the Macdonald index
for odd $N$. From the index, we argue that certain short-multiplet which can
appear in the OPE of the stress-energy tensor is absent in the $(A_1, A_{2n})$
theory. We also discuss the mixed Schur indices for the N=1 class S theories
with irregular punctures.
