Basis partitions (Q1377732)

Let \(n= \pi_1+\pi_2+\cdots+ \pi_l\), \(\pi_1\geq \pi_2\geq\cdots\geq \pi_l\), \(\pi_i\) a positive integer. Then the partition \(\pi= (\pi_1,\dots, \pi_l)\) has weight \(n\). \textit{H. Gupta} [Fibonacci Q. 16, 548-552 (1978; Zbl 0399.10017)] defined a basis partition, that, in the class of all partitions with fixed rank vector, has minimum weight. The rank vector of \(\pi\) is \([\pi_1- \pi_1',\dots, \pi_{d(\pi)}- \pi_{d(\pi)}']\), where \(\pi'= (\pi_1',\dots, \pi_m')\) is the conjugate partition and \(d(\pi)\) is the size of the Durfee square. For these partitions the authors derive a recurrence relation, a generating function, some identities relating basis partitions to other families of partitions, and a new characterization of the basis partitions.