Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Regular structures of an intractable enumeration problem: a diagonal recurrence relation of monomer-polymer coverings on two-dimensional rectangular lattices

Yong Kong

Abstract

The enumeration of polymer coverings on two-dimensional rectangular lattices is considered as "intractable". We prove that the number of coverings of $s$ polymer satisfies a simple recurrence relation $ \sum_{i=0}^{2s} (-1)^i \binom{2s}{i} a_{n-i, m-i} = 2^s {(2s)!} / {s!} $ on a $n \times m$ rectangular lattice with open boundary conditions in both directions.

Regular structures of an intractable enumeration problem: a diagonal recurrence relation of monomer-polymer coverings on two-dimensional rectangular lattices

Abstract

The enumeration of polymer coverings on two-dimensional rectangular lattices is considered as "intractable". We prove that the number of coverings of polymer satisfies a simple recurrence relation on a rectangular lattice with open boundary conditions in both directions.

Paper Structure

This paper contains 28 sections, 51 theorems, 216 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A two-dimensional sequence
  3. Proof of the main theorem
  4. Outline of the proof
  5. The two sets of recurrence identities: $\mathcal{P}_1$ and $\mathcal{P}_2$
  6. The set of recurrence identities in $\mathcal{P}_1$
  7. The set of recurrence identities in $\mathcal{P}_2$
  8. The first quadrant $Q_1$
  9. The third quadrant $Q_3$
  10. Summary
  11. The contributions of $h_{2s-i, s-j}^{(1)}$ cancels the vertical recurrences of $\mathcal{P}_1$
  12. The contributions of $h_{2s-i,s-j}^{(2)}$ cancel the horizontal recurrences of $\mathcal{P}_1$
  13. The contributions of $h_{2s-i, s-j'}^{(3)}$ vanish
  14. Outline of the proof
  15. Transforming the inner sum
  16. ...and 13 more sections

Key Result

Theorem 1

The number of configurations of $s$$k$-mers on a $n \times m$ lattice with open boundary conditions in both directions satisfies the following recurrence for $n \ge (k+1)s$ and $m \ge (k+1)s$.

Figures (4)

  • Figure 1: (Color online) The set of recurrence identities in $\mathcal{P}_1$ for $s=4$. The arrows labeled in red are for the vertical recurrences; those in blue for the horizontal recurrences. The numbers are the coefficients $b_{ij}$ (weights) for the recurrences.
  • Figure 2: (Color online) The set of recurrence identities in $\mathcal{P}_1$ for $s=5$.
  • Figure 3: (Color online) The set of recurrence identities in $\mathcal{P}_2$ for $s=4$. The arrows labeled in red are for the vertical recurrences; those in blue for the horizontal recurrences. The numbers are the coefficients $\bar{h}_{i,j}$ (weights) for the recurrences.
  • Figure 4: (Color online) The set of recurrence identities in $\mathcal{P}_2$ for $s=5$.

Theorems & Definitions (88)

  • Theorem 1
  • Corollary 1.1
  • proof
  • Lemma 1: Recurrence on a lattice strip
  • Remark 1
  • Remark 2
  • Definition 1
  • Lemma 2: Recursive relation of the generating function
  • proof
  • Proposition 1: Explicit formula for $H_{i}$
  • ...and 78 more