Counting singular curves with tangencies

We obtain a recursive formula for the characteristic number of degree $d$ curves in $\mathbb{P}^2$ with prescribed singularities (of type $A_k$) that are tangent to a given line. The formula is in terms of the characteristic number of curves with exactly those singularities. Combined with the results of S. Basu and R. Mukherjee, this gives us a complete formula for the characteristic number of curves with $\delta$-nodes and one singularity of type $A_k$, tangent to a given line, provided $\delta + k \leq 8$. We use a topological method, namely the method of dynamic intersections to compute the degenerate contribution to the Euler class. Till codimension eight, we verify that our numbers are logically consistent with those computed earlier by Caporaso-Harris. We also make a non trivial low degree check to verify our formula for the number of cuspidal cubics tangent to a given line, using a result of Kazarian.