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Hankel Determinants of convoluted Catalan numbers and nonintersecting lattice paths: A bijective proof of Cigler's Conjecture

Markus Fulmek

Abstract

In recent preprints, Cigler considered certain Hankel determinants of convoluted Catalan numbers and conjectured identities for these determinants. In this note, we shall give a bijective proof of Cigler's Conjecture by interpreting determinants as generating functions of nonintersecting lattice paths: this proof employs the reflection principle, the Lindström-Gessel-Viennot-method and a certain construction involving reflections and overlays of nonintersecting lattice paths. Shortly after this bijective proof was presented here, Cigler provided a shorter proof based on earlier results.

Hankel Determinants of convoluted Catalan numbers and nonintersecting lattice paths: A bijective proof of Cigler's Conjecture

Abstract

In recent preprints, Cigler considered certain Hankel determinants of convoluted Catalan numbers and conjectured identities for these determinants. In this note, we shall give a bijective proof of Cigler's Conjecture by interpreting determinants as generating functions of nonintersecting lattice paths: this proof employs the reflection principle, the Lindström-Gessel-Viennot-method and a certain construction involving reflections and overlays of nonintersecting lattice paths. Shortly after this bijective proof was presented here, Cigler provided a shorter proof based on earlier results.
Paper Structure (40 sections, 6 theorems, 32 equations)

This paper contains 40 sections, 6 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminary considerations
  3. Binomial coefficients and lattice paths
  4. Determinants interpreted as generating functions
  5. The Lindström--Gessel--Viennot method
  6. Enforced segments for survivors
  7. Positions of enforced initial points
  8. Even identities
  9. Odd identities
  10. Even and odd identities
  11. A first consequence
  12. A warmup--exercise: Cigler's Conjecture for $k=1$
  13. Bijective proof of Cigler's Conjecture for $k>1$
  14. Encoding permutations of survivors
  15. Relations between signs of admissible permutations and their codes
  16. ...and 25 more sections

Key Result

Theorem 1

Let $m,k \in{\mathbb N}$, with $m,k>0$. Then for $K=2k$ we have the even identities and for all $n\in{\mathbb N}$ Moreover, for $K=2k-1$ we have the odd identities and for all $n\in{\mathbb N}$

Theorems & Definitions (11)

  • Theorem 1: Cigler's Conjecture
  • proof : Proof of Cigler's Conjecture for special case $k=1$
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 1 more