  Strong stability preserving (SSP) coefficients govern the maximally allowable
step-size at which positivity or contractivity preservation of integration
methods for initial value problems is guaranteed. In this paper, we show that
the task of computing linear SSP coefficients of explicit one-step methods is,
to a certain extent, equivalent to the problem of characterizing positive
quadratures with integer nodes with respect to Poisson-Charlier measures. Using
this equivalence, we provide sharp upper and lower bounds for the optimal
linear SSP coefficients in terms of the zeros of generalized Laguerre
orthogonal polynomials. This in particular provides us with a sharp upper bound
for the optimal SSP coefficients of explicit Runge-Kutta methods. Also based on
this equivalence, we propose a highly efficient and stable algorithm for
computing these coefficients, and their associated optimal linear SSP methods,
based on adaptive spectral transformations of Poisson-Charlier measures. The
algorithm possesses the remarkable property that its complexity depends only on
the order of the method and thus is independent of the number of stages. Our
results are achieved by adapting and extending an ingenious technique by
Bernstein in his seminal work on absolutely monotonic functions. Moreover, the
techniques introduced in this work can be adapted to solve the integer
quadrature problem for any positive discrete multi-parametric measure supported
on $\mathbb{N}$ under some mild conditions on the zeros of the associated
orthogonal polynomials.
