The distributivity numbers of finite products of P(ω)/fin

Generalizing [ShSp], for every n < ω we construct a ZFC-model where Heng hooktop sign(n) the distributivity number of r.o(P(ω)/fin)ⁿ, is greater than Heng hooktop sign(n + 1) This answers an old problem of Balcar, Pelant and Simon (see [BaPeSi]). We also show that bot Laver and Miller forcings collapse the continuum to Heng hooktop sign(n) for every n < ω, hence by the first result, consistently they collapse it below Heng hooktop sing(n).