Identities and generating functions for certain classes of \(F\)-partitions. (Q2715970)

An \(F\)-partition (\(F\) is for Frobenius) of \(v\) is a decomposition \(v=r+a_1+\cdots +a_r+b_1+\cdots +b_r\) where \(a_1\geq a_2\geq \ldots \geq a_r\geq 0\) and the same holds for the \(b_i\)s. An \((n+t)\)-color partition, \(t\geq 0\), is the standard partition in which part \(n\geq 0\) may come in \(n+t\) colors distinguished by the subscripts \(n_1,\dots ,n_{n+t}\). Zeros as parts are admitted only if \(t>0\) and they are not allowed to repeat. The author proves four identities saying that certain sets \(A\) of \(F\)-partitions and sets \(B\) of \((n+t)\)-color partitions are equinumerous. For example, in Theorem 3 (where \(k\) is a parameter) \(A\) is the set of \(F\)-partitions of \(v\) in which \(a_i\neq 1\), \(a_i\leq 2+b_i\), \(a_i>(k+7)/2+b_{i+1}\), and \(a_r>0\Rightarrow a_r>(k+3)/2\), and \(B\) is the set of \((n+2)\)-color partitions of \(v\) in which if \(m_i\) is not the smallest part then \(m-i>0\), each part has the same parity as its subscript, \(m-n-i-j>k\) whenever \(m_i,n_j\) are parts with \(m\geq n\), and the smallest part has the form \(i_{i+2}\). The proof of Theorem 3 is bijective; the other three proofs are only sketched. Formulas for generating functions of numbers of the \(F\)-partitions considered are given as well as various identities following from particular cases of the four theorems (\(k\) is set to be \(1\) or \(-1\)).