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Two New Integer Sequences Related to Crossroads and Catalan Numbers

Julien Rouyer, Alain Ninet

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.

Two New Integer Sequences Related to Crossroads and Catalan Numbers

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.
Paper Structure (5 sections, 11 theorems, 14 equations, 7 figures, 1 table)

This paper contains 5 sections, 11 theorems, 14 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Standard road intersection
  3. Properties
  4. Known values and conjectures
  5. Acknowledgements

Key Result

Lemma 6

Let $C_n$ (for $n\geq 0$) denote the number of noncrossing partitions of $[n]$. Then, we have $C_n=L_n+M_n$.

Figures (7)

  • Figure 1: A Standard Road Intersection of size $n=4$.
  • Figure 2: A bipartite graph associated with the intersection represented in Figure \ref{['fig:inter']}, with an example of absolute MSL corresponding to the noncrossing partition $\{\,\{1,2,3\},\{4\}\,\}$.
  • Figure 3: A bipartite graph associated with the intersection represented in Figure \ref{['fig:inter']}, with an example of a nonabsolute MSL corresponding to the noncrossing partition $\{\,\{1,2\},\{3\},\{4\}\,\}$.
  • Figure 4: Simplified graph of Figure \ref{['fig:cercle']}, showing explicitly the noncrossing lonely singles partition $\{\,\{1,2,3\},\{4\}\,\}$.
  • Figure 5: Simplified graph of Figure \ref{['fig:cerclenonmax']}, showing explicitly the noncrossing marriageable singles partition $\{\,\{1,2\},\{3\},\{4\}\,\}$.
  • ...and 2 more figures

Theorems & Definitions (34)

  • Definition 1
  • Remark 4
  • Lemma 6
  • proof
  • Remark 7
  • Remark 8
  • Definition 9
  • Definition 10
  • Definition 11
  • Definition 12
  • ...and 24 more