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 $q\ge 1$, 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.