Inversion of Incidence Mappings

Denote by H(t, q), t ≤ q, the incidence matrix (with respect to inclusion) of the t–sets versus the q–sets of the n–set {1, 2, . . . , n}. This matrix is considered as a linear map of Q–vector spaces Cq(n) −→ Ct(n), where Cs(n) is the Q–vector space having the s–sets as a basis (s ≤ n). As a basic tool, we introduce a connection of the vector spaces to a graded Q–algebra (which is at the same time an Artinian local ring). We define mappings 4 and X of this algebra of degree −1 and 1, respectively. These two mappings correspond up to a scalar factor to the linear mappings H(s − 1, s) and H(s− 1, s)T , respectively. Then, a relation between the algebra maps 4 and X is established. This relation allows to rewrite a term4βXα with α, β non–negative integers (subject to some restrictions) as a sum β ∑ k=0 ( β k ) ·Xα−k ·4β−k (up to some scalar factors). As a main result of this relation surjectivity of the map4q−t (related to H(t, q) up to a scalar factor) is proved under the assumption ( n t ) ≤ ( n q ) . Moreover, a right inverse for the matrix H(t, q) is given explicitely. This result is exploited to give an inverse of the (square) incidence matrix H(t, q) in the case t = n− q. These results extend some work done by J.B. Graver and W.B. Jurkat.