  The paper addresses a conjecture of Shapiro and Tater on the similarity
between two sets of points in the complex plane; on one side is the values of
$t\in \mathbb{C}$ for which the spectrum of the quartic anharmonic oscillator
in the complex plane $$\frac{{\rm d}^2 y}{{\rm d}x^2} - ( x^4 + tx^2 + 2Jx )y =
\Lambda y, $$ with certain boundary conditions, has repeated eigenvalues. On
the other side is the set of zeroes of the Vorob'ev-Yablonskii polynomials,
i.e. the poles of rational solutions of the second Painlev\'e equation. Along
the way, we indicate a surprising and deep connection between the anharmonic
oscillator problem and certain degenerate orthogonal polynomials.
