Derangements of a deck of cards, where ranks are equal

This answer by joriki to an earlier question is easily modified to provide an answer to this one. This is a generalized derangement problem, and in the notation of that answer we have $r=13$ and $n_i=4$ for $i=1,\ldots,13$. The $4$th Laguerre polynomial is

$$L_4(x)=\frac1{24}\left(x^4-16x^3+72x^2-96x+24\right)\;,$$

and the $4$ cards of each rank are distinguishable, so there are

$$(4!)^{13}\int_0^\infty\big(L_4(x)\big)^{13}e^{-x}\,dx=\int_0^\infty\left(x^4-16x^3+72x^2-96x+24\right)^{13}e^{-x}\,dx$$

permutations that do not yield a match. According to WolframAlpha this yields a probability of

$$\frac{4,610,507,544,750,288,132,457,667,562,311,567,997,623,087,869}{52!}\approx0.016233\;.$$