A Lemma and a Conjecture on the Cost of Rearrangements

TL;DR

The paper investigates the minimal total cost required to rearrange a mixed stack of black and white books into sorted order using elementary transpositions, providing a conjecture and a lemma related to this problem.

Contribution

It introduces a new lemma and formulates a conjecture to establish lower bounds on the cost of rearrangements in a combinatorial sorting problem.

Findings

Proposes a lower bound on rearrangement costs.

Introduces a lemma relevant to the cost analysis.

Presents a conjecture guiding future research.

Abstract

Consider a stack of books, containing both white and black books. Suppose that we want to sort them out, putting the white books on the right, and the black books on the left (fig.~1). This will be done by a finite sequence of elementary transpositions. In other words, if we have a stack of all black books of length $a$ followed by a stack of all white books of length $b$, we are allowed to reverse their order at the cost of $a+b$. We are interested in a lower bound on the total cost of the rearrangement.