Basis partitions. (English) Zbl 0901.05007
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 \(n\). 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.
Reviewer: G.L.Alexanderson (Santa Clara)
MSC:
05A17 | Combinatorial aspects of partitions of integers |
05A15 | Exact enumeration problems, generating functions |
Citations:
Zbl 0399.10017Online Encyclopedia of Integer Sequences:
Number of basis partitions (or basic partitions) of n.Triangle T(n,k) giving number of basis partitions of n with a Durfee square of order k (n >= 0, 0 <= k <= n).
References:
[1] | Atkin, A. O.L., A note on ranks and conjugacy of partitions, Quart. J. Math., 17, 335-338 (1966) · Zbl 0144.25203 |
[2] | Gupta, H., The rank-vector of a partition, Fibonacci Quart., 16, 548-552 (1978) · Zbl 0399.10017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.