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We elaborate on a recently conjectured relation of Painlev\'e transcendentsand 2D CFT. General solutions of Painlev\'e VI, V and III are expressed interms of $c=1$ conformal blocks and their irregular limits, AGT-related toinstanton partition functions in $\mathcal{N}=2$ supersymmetric gauge theorieswith $N_f=0,1,2,3,4$. Resulting combinatorial series representations ofPainlev\'e functions provide an efficient tool for their numerical computationat finite values of the argument. The series involve sums over bipartitionswhich in the simplest cases coincide with Gessel expansions of certain Toeplitzdeterminants. Considered applications include Fredholm determinants ofclassical integrable kernels, scaled gap probability in the bulk of the GUE,and all-order conformal perturbation theory expansions of correlation functionsin the sine-Gordon field theory at the free-fermion point.

How instanton combinatorics solves Painlev\'e VI, V and III's