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Abstract

We study the correlators of irregular vertex operators in two-dimensional
conformal field theory (CFT) in order to propose an exact analytic formula for
calculating numbers of partitions, that is:
1) for given $N,k$, finding the total number $\lambda(N|k)$ of length $k$
partitions of $N$: $N=n_1+...+n_k;0<n_1\leq{n_2}...\leq{n_k}$.
2) finding the total number $\lambda(N)=\sum_{k=1}^N\lambda(N|k)$ of
partitions of a natural number $N$
We propose an exact analytic expression for $\lambda(N|k)$ by relating
two-point short-distance correlation functions of irregular vertex operators in
$c=1$ conformal field theory ( the form of the operators is established in this
paper): with the first correlator counting the partitions in the upper
half-plane and the second one obtained from the first correlator by conformal
transformations of the form $f(z)=h(z)e^{-{i\over{z}}}$ where $h(z)$ is regular
and non-vanishing at $z=0$. The final formula for $\lambda(N|k)$ is given in
terms of regularized ($\epsilon$-ordered) finite series in the generalized
higher-derivative Schwarzians and incomplete Bell polynomials of the above
conformal transformation at $z=i\epsilon$ ($\epsilon\rightarrow{0}$)