A Note on Partitions and Compositions Defined by Inequalities

for some positive integers b1, . . . , bn. Specifically, it was shown that if C is an upper triangular matrix with all ci,i = 1, and if B = C−1 has nonnegative entries, then FC(q) is given by (2), where bj is the sum of the entries in column j of B = [bi,j ]. Moreover, it was shown that this provides a natural bijection between the “partitions” of N into parts from the multiset {b1, . . . , bn} and the compositions of N in SC given by: b1 1 b m2 2 · · · bn n ↔ (λ1, λ2, . . . , λn), (3) ∗Research supported in part by NSF grants DMS-0300034 and INT-0230800