On the number of partitions of \(n\) into exactly \(m\) parts whose even parts are distinct (Q6614173)

A partition of a positive integer \(n\) is a finite sequence of positive integers \(\lambda=\left(\lambda_1, \lambda_2, \ldots, \lambda_m\right)\) such that\N\[\N\lambda_1+\lambda_2+ \ldots+ \lambda_m=n.\N\]\NThe \(\lambda_i\)'s are called the parts of the partition. The number of parts is unrestricted, repetition is allowed, and the order of the parts is not taken into account (nevertheless, the usual assumption is that \(\lambda_1\geq \lambda_2\geq \ldots\geq \lambda_m)\).\N\NLet \(ped(n)\) be the number of partitions of n whose even parts are distinct and whose odd parts are unrestricted. The function \(ped(n)\) has been studied by many researchers and finding a new formula for the numbers of partitions of \(n\) with distinct even parts is a problem of current interest.\N\NFor a positive integer \(m\), let \(ped(n,m)\) be the number of all possible partitions of the number \(n\) into exactly \(m\) parts whose even parts are distinct and whose odd parts are unrestricted. In this paper, the authors presented new recurrence formulas for \(ped(n,m)\):\N\[\Nped(n,m)=ped(n-1,m-1)+ped(n-2m, m-1)+ped(n-2m,n),\N\]\Nwhere \(n\geq 1\) and \(1\leq m\leq n\). Also, explicit formulas for \(ped(n,m)\), when \(m=2,3\) and \(m=4\) were established.