A Runlength Operator on Partitions of Integers, Applied to Inventory Chains

Abstract

The inventory of a \(2 \times m\) array \(A = A(i,j)\) consisting of \(n\) not necessarily distinct positive integers \(\mathbb{I}(2,j)\) is the \(2 \times n\) array \(\mathbb{I}(A) = \mathbb{I}(i,j)\), where \(\mathbb{I}(i,j)\) is the number of occurrences of \(\mathbb{I}(1,j)\) in \(A\). Define \(\mathbb{I}^q(A) = I(\mathbb{I}^{q-1}(A))\) for \(q \geq 1\), with \(\mathbb{I}^0(A) = A\). For every \(A\), the chain \(\{\mathbb{I}^q(A)\}\) of inventories is eventually periodic, with period \(1, 2\), or \(3\). The proof depends on runlengths of partitions of integers. A final section is devoted to an open question about cumulative inventory chains.