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Bijections and congruences involving lattice paths and integer compositions

Manosij Ghosh Dastidar, Michael Wallner

Abstract

We prove new bijections between different variants of Dyck paths and integer compositions, which give combinatorial explanations of their simple counting formula $4^{n-1}$. These give relations between different statistics, such as the number of crossings of the $x$-axis in classes of Dyck bridges or the distribution of peaks in classes of Dyck paths, and furthermore relate them with $k$- and $g$-compositions. These allow us to find and prove congruence results for Dyck paths and parity results for compositions. Our investigation uncovers unexpected connections to mock theta functions, Hardinian arrays, little Schröder paths, Fibonacci numbers, and irreducible pairs of compositions, offering new insights into the structures of paths, partitions and compositions.

Bijections and congruences involving lattice paths and integer compositions

Abstract

We prove new bijections between different variants of Dyck paths and integer compositions, which give combinatorial explanations of their simple counting formula . These give relations between different statistics, such as the number of crossings of the -axis in classes of Dyck bridges or the distribution of peaks in classes of Dyck paths, and furthermore relate them with - and -compositions. These allow us to find and prove congruence results for Dyck paths and parity results for compositions. Our investigation uncovers unexpected connections to mock theta functions, Hardinian arrays, little Schröder paths, Fibonacci numbers, and irreducible pairs of compositions, offering new insights into the structures of paths, partitions and compositions.
Paper Structure (12 sections, 28 theorems, 38 equations, 5 figures)

This paper contains 12 sections, 28 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Bijections between compositiosn and lattice paths
  3. Bijections involving compositions and arrays
  4. Bijections involving peaks and crossings in paths
  5. Compositions and paths in a strip
  6. Congruences and Peak Profiles for Paths
  7. Little and large Schröder paths
  8. Dyck paths with exactly r peaks per reached level
  9. Congruences for the number of specific steps in paths
  10. Restrictions on peaks and parts
  11. Restricted summits
  12. First, Greatest and Smallest Parts

Key Result

Proposition 2.1

There exists a natural bijection between pairs of compositions of $n$ and Dyck walks of length $2n-2$.

Figures (5)

  • Figure 1: Bijections proved in this paper of classes of paths and compositions, all enumerated by $4^{n-1}$.
  • Figure 2: An equivalent representation of a Hardinian array with $r=1$ as a triangle of convex compositions (obeying rules analogous to Definition \ref{['def:Hardinian']}).
  • Figure 3: The Hardinian array from Figure \ref{['fig:Hardin2ConvexComp']} is in bijection with the pair of compositions $((1,2,2),(1,1,1,2))$ with equal first part $1$; see Proposition \ref{['prop:HardinBijection']}.
  • Figure 4: Dyck path with a marked peak (red dot) at height 6 and image under bijection from Theorem \ref{['theo:Dyckpeak']} given by a Dyck bridge starting with a $\mathtt d$ step and 5=6-1 crossings (red dots). The black steps are used in the last-passage (resp., first-passage) decomposition in the proof.
  • Figure 5: Last-passage decomposition of a Dyck bridge used in Lemmas \ref{['lem:DyckbridgeFirstPeak']} and \ref{['lem:Dyckbridgeirreduciblepairs']}

Theorems & Definitions (75)

  • Proposition 2.1
  • proof
  • Example 2.2
  • Proposition 2.3
  • proof
  • Example 2.4
  • Proposition 2.5: HopkinsOuvry2021Multicompositions
  • proof
  • Definition 2.6: Hardinian Arrays
  • Proposition 2.7
  • ...and 65 more