Left-inversion of combinatorial sums

The inversion of combinatorial sums is a fundamental problem in algebraic combinatorics. Some combinatorial sums, such as a_n = Σ_kd_n,kb_k, cannot be inverted in terms of the orthogonality relation because the infinite, lower triangular array P = {d_n,k}'s diagonal elements are equal to zero (except d_0,0). Despite this, we can find a left-inverse ̄P such that PP̄ = I and therefore are able to left-invert the original combinatorial sum, and thus obtain b_n = Σ_kd̄_n,ka_k.