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The first measurements of the scattering parameters of $\Lambda$K pairs inall three charge combinations ($\Lambda$K$^{+}$, $\Lambda$K$^{-}$, and$\Lambda\mathrm{K^{0}_{S}}$) are presented. The results are achieved through afemtoscopic analysis of $\Lambda$K correlations in Pb-Pb collisions at$\sqrt{s_{\mathrm{NN}}}$ = 2.76 TeV recorded by ALICE at the LHC. Thefemtoscopic correlations result from strong final-state interactions, and arefit with a parametrization allowing for both the characterization of the pairemission source and the measurement of the scattering parameters for theparticle pairs. Extensive studies with the THERMINATOR 2 event generatorprovide a good description of the non-femtoscopic background, which resultsmainly from collective effects, with unprecedented precision. Furthermore,together with HIJING simulations, this model is used to account forcontributions from residual correlations induced by feed-down from particledecays. The extracted scattering parameters indicate that the strong force isrepulsive in the $\Lambda\rm{K}^{+}$ interaction and attractive in the$\Lambda\rm{K}^{-}$ interaction. The data hint that the and$\Lambda\rm{K}^{0}_{S}$ interaction is attractive, however the uncertainty ofthe result does not permit such a decisive conclusion. The results suggest aneffect arising either from different quark-antiquark interactions between thepairs ($\rm s\overline{s}$ in $\Lambda$K$^{+}$ and $\rm u\overline{u}$ in$\Lambda$K$^{-}$) or from different net strangeness for each system (S = 0 for$\Lambda$K$^{+}$, and S = $-2$ for $\Lambda$K$^{-}$). Finally, the $\Lambda$Ksystems exhibit source radii larger than expected from extrapolation fromidentical particle femtoscopic studies. This effect is interpreted as resultingfrom the separation in space-time of the single-particle $\Lambda$ and K sourcedistributions.

$\Lambda\rm{K}$ femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV