Some wonderful conjectures (but almost no theorems) at the boundary between analysis, combinatorics and probability

I discuss some remarkable properties of the entire function $F(x,y) = \sum\limits_{n=0}^\infty {x^n \over n!} \, y^{n(n-1)/2}$, the polynomials $P_N(x,w) = \sum\limits_{n=0}^N \binom{N}{n} x^n w^{n(N-n)}$, and the generating polynomials $C_n(v) = \sum\limits_{m=n-1}^{n(n-1)/2} c_{n,m} v^m$ of connected graphs. These objects arise in several contexts in graph theory and statistical physics.