  We introduce thagomizer matroids and compute the Kazhdan-Lusztig polynomial
of a rank $n+1$ thagomizer matroid by showing that the coefficient of $t^k$ is
equal to the number of Dyck paths of semilength $n$ with $k$ long ascents. We
also give a conjecture for the $S_n$-equivariant Kazhdan-Lusztig polynomial of
a thagomizer matroid.
