An Analytic Formula for Numbers of Restricted Partitions from Conformal Field Theory

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 $\lambda (N\mathrm{\mid }k)$ of length $k$ partitions of $N$: $N=n_1+\mathrm{.}\mathrm{.}\mathrm{.}+n_k;0<n_1\le n_2\mathrm{.}\mathrm{.}\mathrm{.}\le n_k$.
2. finding the total number $\lambda (N)={\sum }_{k=1}^N\lambda (N\mathrm{\mid }k)$ of partitions of a natural number $N$ We propose an exact analytic expression for $\lambda (N\mathrm{\mid }k)$ 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 $f(z)=h(z)e^{-\frac{i}{z}}$ where $h(z)$ is regular and non-vanishing at $z=0$. The final formula for $\lambda (N\mathrm{\mid }k)$ is given in terms of regularized ($ϵ$-ordered) finite series in the generalized higher-derivative Schwarzians and incomplete Bell polynomials of the above conformal transformation at $z=iϵ$ ($ϵ\to 0$)