Enumeration of Payphone Permutations

TL;DR

The paper studies a privacy-driven permutation model where $n$ people occupy payphones in a row to maximize distance from occupied phones. It develops an interval-evolution framework with $S(n)$ and derives an explicit formula for the multiplicities $#_k S(n)$ via a quasi-binary representation, enabling exact enumeration of several permutation types. Using multiplicity-based functions $F$, $G$, and $H$, the authors give closed-form enumeration formulas for type (C2), (C1), (P2), (P1) and variants (P3)–(P5), linking the counts to finite multisets of interval lengths. The work yields numerous counted terms, reveals non-monotonic patterns in some sequences, and connects to OEIS while offering a general combinatorial approach to parking-like problems and related permutation enumeration.

Abstract

The desire for privacy significantly impacts various aspects of social behavior as illustrated by people's tendency to seek out the most secluded spot when multiple options are available. In particular, this can be seen at rows of payphones, where people tend to occupy an available payphone that is most distant from those already occupied. Assuming that there are n payphones in a row and that n people occupy payphones one after another as privately as possible, the resulting assignment of people to payphones defines a permutation, which we will refer to as a payphone permutation. In the present study, we consider different variations of payphone permutations and enumerate them.

Figures (2)

• Figure 1: A row of payphones. (pxfuel.com)
• Figure 2: The top part of the tree $T$ composed of nodes with labels from $1$ to $63$.

Theorems & Definitions (18)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof