Logarithmic integrals, zeta values, and tiered binomial coefficients

We study logarithmic integrals of the form $$\int _0^1 x^i\ln ^n(x)\ln ^m(1-x)dx$$ . They are expressed as a rational linear combination certain numbers $$(n,m)_{i}$$ , which we call tiered binomial coefficients, and products zeta values $$\zeta (2)$$ (3)$$ ,.... Various properties coefficients established. involve, amongst others, transform, truncated multiple star values, well special functions. present extensions to generalized Nielsen polylogarithms. As an application revisit limit law number comparisons Quicksort algorithm: reprove that moments polynomials in values. Properties cumulants also discussed.