On pairs of partitions with steadily decreasing parts
If \(n\) and \(m\) are non-negative integers, let \(\alpha= (\alpha_1, \alpha_2, \dots,\alpha_r)\) be a partition of \(n\), while \(\beta= (\beta_1, \beta_2, \dots,\beta_s)\) is a partition of \(m\). If the pair of partitions \((\alpha,\beta)\) satisfies the conditions: \(\min(\alpha_i, \beta_i) \geq \max(\alpha_{i+1}, \beta_{i+1})\) for all \(i\), then \((\alpha,\beta)\) is said to have steadily decreasing parts. Such pairs of partitions have been previously been studied by Carlitz and by Andrews. The author represents such pairs of partitions graphically via a modified Young diagram. Her main result concerns \(R(n,m)\), the set of all pairs \((\alpha,\beta)\) of partitions of \(n,m\) with steadily decreasing parts such that in addition (1) the \(\alpha_i\) are distinct; (2) the \(\beta_j\) are distinct; (3) \(\alpha_i\neq\beta_i\) for all \(i\). For non-negative integers \(k,n,m\), let \(R(k)=\cup \{R(n,m): n+m=k\}\). Let \(D,O\) represent, respectively, the set of all partitions of non-negative integers into distinct parts, odd parts. Let \(Q(k)\) denote the set of all pairs \(\lambda,\mu\) of partitions such that \(\lambda\in D\cap O\), \(\mu\in O\), and \(|\lambda |+|\mu|=k\). Then \(|R(k)|=|Q(k)|\). The reader is cautioned that since the possibilities \(n,m=0\) are not excluded, some of the partitions under consideration are empty.
scientific article; zbMATH DE number 1888407