Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

About maximal antichains in a product of two chains:A catch-all note

Denis Bouyssou, Thierry Marchant, Marc Pirlot

Abstract

We establish one-to-one correspondences between maximal antichains in products of two finite linear orders and other mathematical objects, such as certain alignments of two strings, walks on a grid, lattice paths, words of two or three letters. Leaning on these correspondences, we gather what is known about the number of maximal antichains in products of two finite linear orders and we establish some new results.

About maximal antichains in a product of two chains:A catch-all note

Abstract

We establish one-to-one correspondences between maximal antichains in products of two finite linear orders and other mathematical objects, such as certain alignments of two strings, walks on a grid, lattice paths, words of two or three letters. Leaning on these correspondences, we gather what is known about the number of maximal antichains in products of two finite linear orders and we establish some new results.

Paper Structure

This paper contains 31 sections, 13 theorems, 24 equations, 4 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Dominance orders
  3. Chains and antichains
  4. Correspondences
  5. Antichains and strict chains
  6. Noses of a step matrix and augmentation matrix
  7. Augmentation matrix of an antichain
  8. Step matrices and augmentation matrix for strict chains
  9. Alignments of two strings
  10. Alignments and strict chains
  11. Alignments without alternate skips and maximal strict chains
  12. Words using an alphabet of three letters
  13. Alignments and words
  14. Strict chains and words via grid lines
  15. Antichains and words via grid lines
  16. ...and 16 more sections

Key Result

Proposition 1

There is a bijection between strict chains and antichains in $[m_1]\times[m_2]\xspace$.

Figures (4)

  • Figure 1: Matrix of strict chain $\{(2,3), (4,5)\}$ represented in Table \ref{['ta:augmMatrixAlign2']}. The boldface grid line separates the SW region determined by the chain from the other cells.
  • Figure 2: Matrix of antichain $\{(2,4), (4,2)\}$ in $[5]\times[6]$. The boldface grid line separates the SE region determined by the chain from the other cells.
  • Figure 3: A grid walk from $(0,0)$ to $(6,5)$. The strict chain $\{(1,1), (3,3), (4,4), (6,5)\}$ is not maximal because it has a $v'h'$ subsequence (namely $(1,1), (1,2), (2,2)$) disjoint from all $h'v'$ subsequences.
  • Figure 4: A grid walk from $(0,5)$ to $(6,0)$. The antichain $\{(1,4), (3,2), (4,1), (6,0)\}$ is not maximal because it has a $v'h'$ subsequence

Theorems & Definitions (20)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Remark 1
  • Proposition 4
  • Proposition 5
  • Remark 2
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • ...and 10 more