Two New Integer Sequences Related to Crossroads and Catalan Numbers

TL;DR

This paper introduces two new integer sequences related to noncrossing partitions, exploring their properties and applications in counting crossing road intersections, thereby extending combinatorial enumeration theory.

Contribution

It defines and analyzes the lonely singles and marriageable singles sequences, connecting them to Catalan numbers and their combinatorial interpretations.

Findings

Sequences are explicitly computed for the first 14 terms.

Properties and relationships with Catalan numbers are established.

Applications in counting crossing road intersections are demonstrated.

Abstract

The lonely singles sequence represents the number of noncrossing partitions of the finite set {1,. .. , n} in which no pair of singletons {i} and {j} can be merged into the pair {i, j} so that the partition stays noncrossing. The marriageable singles sequence represents the number of all the other noncrossing partitions and is the difference between the Catalan numbers sequence and the lonely singles sequence. The 14 first terms of these sequences are given, as well as some of their properties. These sequences appear when one wants to count the number of ways to cross simultaneously certain road intersections.