We use Monte-Carlo simulations, combined with homogeneously determined age and mass distributions based on multi-wavelength photometry, to constrain the cluster formation history and the rate of bound cluster disruption in the Large Magellanic Cloud (LMC) cluster system. We evolve synthetic star cluster systems formed with a power-law initial cluster mass function (ICMF) of spectral index $alpha =-2$ assuming different cluster disruption time-scales. For each of these disruption time-scales we derive the corresponding cluster formation rate (CFR) required to reproduce the observed cluster age distribution. We then compare, in a Poissonian $chi^2$ sense, model mass distributions and model two-dimensional distributions in log(mass) vs. log(age) space of the detected surviving clusters to the observations. Because of the bright detection limit ($M_V^{rm lim} simeq -4.7$ mag) above which the observed cluster sample is complete, one cannot constrain the characteristic disruption time-scale for a $10^4$ M$_odot$ cluster, $t_4^{rm dis}$ (where the disruption time-scale depends on cluster mass as $t_{rm dis} = t_4^{rm dis} (M_{rm cl} / 10^4 {rm M}_odot)^0.62$), to better than $t_4^{rm dis} ge 1$ Gyr. We conclude that the CFR has increased from 0.3 clusters Myr$^{-1}$ 5 Gyr ago, to a present rate of $(20-30)$ clusters Myr$^{-1}$. For older ages the derived CFR depends sensitively on our assumption of the underlying CMF shape. If we assume a universal Gaussian ICMF, then the CFR has increased steadily over a Hubble time from $sim 1$ cluster Gyr$^{-1}$ 15 Gyr ago to its present value. If the ICMF has always been a power law with a slope close to $alpha=-2$, the CFR exhibits a minimum some 5 Gyr ago, which we tentatively identify with the well-known age gap in the LMCs cluster age distribution.
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