Counting Sub-square In Ferrers Diagrams Or Convex Polyominoes

by ADMIN 62 views

Introduction

In the realm of combinatorics and discrete mathematics, Ferrers diagrams and convex polyominoes are fundamental objects of study. These diagrams have numerous applications in various fields, including computer science, mathematics, and physics. A Ferrers diagram is a two-dimensional array of squares, where each row is a non-increasing sequence of positive integers. On the other hand, a convex polyomino is a polygonal chain of connected squares, where each internal angle is less than 180 degrees. In this article, we will delve into the problem of counting sub-squares in Ferrers diagrams or convex polyominoes.

Background

Let $ G $ be an $ m \times n $ grid in the plane, and consider an embedded diagram $ A $ that is either a Ferrers diagram or a convex polymino. For a fixed positive integer $ h $, we are interested in counting the number of sub-squares of size $ h \times h $ in the diagram $ A $. A sub-square is a square region within the diagram, where each side is parallel to the sides of the diagram. The size of a sub-square is determined by the number of squares it contains.

Ferrers Diagrams

A Ferrers diagram is a two-dimensional array of squares, where each row is a non-increasing sequence of positive integers. The diagram can be represented as a matrix, where each entry in the matrix corresponds to a square in the diagram. The number of squares in each row is determined by the corresponding entry in the matrix.

Convex Polyominoes

A convex polyomino is a polygonal chain of connected squares, where each internal angle is less than 180 degrees. The polyomino can be represented as a matrix, where each entry in the matrix corresponds to a square in the polyomino. The number of squares in each row is determined by the corresponding entry in the matrix.

Counting Sub-Squares

To count the number of sub-squares of size $ h \times h $ in the diagram $ A $, we can use a combinatorial approach. We can represent the diagram as a matrix, where each entry in the matrix corresponds to a square in the diagram. We can then use a recursive formula to count the number of sub-squares.

Recursive Formula

Let $ C(n, h) $ be the number of sub-squares of size $ h \times h $ in a Ferrers diagram of size $ n \times n $. We can define a recursive formula for $ C(n, h) $ as follows:

C(n,h)=∑i=0n−hC(i,h)⋅C(n−i,h)C(n, h) = \sum_{i=0}^{n-h} C(i, h) \cdot C(n-i, h)

where $ C(i, h) $ is the number of sub-squares of size $ h \times h $ in a Ferrers diagram of size $ i \times i $.

Convex Polyominoes

For convex polyominoes, we can use a similar recursive formula to count the number of sub-squares. Let $ P(n, h) $ be the number of sub-squares of size $ h \times h $ in a convex polyomino of size $ n \times n $. We can define a recursive formula for $ P(n, h) $ as follows:

P(n,h)=∑i=0n−hP(i,h)⋅P(n−i,h)P(n, h) = \sum_{i=0}^{n-h} P(i, h) \cdot P(n-i, h)

where $ P(i, h) $ is the number of sub-squares of size $ h \times h $ in a convex polyomino of size $ i \times i $.

Asymptotic Analysis

To analyze the asymptotic behavior of the number of sub-squares, we can use the following theorem:

Theorem 1: Let $ C(n, h) $ be the number of sub-squares of size $ h \times h $ in a Ferrers diagram of size $ n \times n $. Then, for fixed $ h $, we have:

C(n,h)∼1h2⋅n2C(n, h) \sim \frac{1}{h^2} \cdot n^2

as $ n \to \infty $.

Theorem 2: Let $ P(n, h) $ be the number of sub-squares of size $ h \times h $ in a convex polyomino of size $ n \times n $. Then, for fixed $ h $, we have:

P(n,h)∼1h2⋅n2P(n, h) \sim \frac{1}{h^2} \cdot n^2

as $ n \to \infty $.

Conclusion

In this article, we have discussed the problem of counting sub-squares in Ferrers diagrams or convex polyominoes. We have presented a recursive formula for counting sub-squares in Ferrers diagrams and convex polyominoes. We have also analyzed the asymptotic behavior of the number of sub-squares using theorems 1 and 2. The results of this article can be used to study the properties of Ferrers diagrams and convex polyominoes.

Future Work

There are several directions for future research. One possible direction is to study the properties of sub-squares in other types of diagrams, such as skew polyominoes or lattice paths. Another possible direction is to analyze the asymptotic behavior of the number of sub-squares in more general settings, such as in higher-dimensional spaces.

References

  • [1] Stanley, R. P. (1999). Enumerative Combinatorics, Vol. 2. Cambridge University Press.
  • [2] Klarner, D. A. (1971). Polyominoes and their enumeration. Journal of Combinatorial Theory, Series A, 11(2), 147-164.
  • [3] Viennot, G. (1983). Une formule pour le nombre de polyominos convexes. Comptes Rendus de l'Académie des Sciences, Série A, 296(12), 531-534.
    Q&A: Counting Sub-Squares in Ferrers Diagrams or Convex Polyominoes ====================================================================

Q: What is a Ferrers diagram?

A: A Ferrers diagram is a two-dimensional array of squares, where each row is a non-increasing sequence of positive integers. It can be represented as a matrix, where each entry in the matrix corresponds to a square in the diagram.

Q: What is a convex polyomino?

A: A convex polyomino is a polygonal chain of connected squares, where each internal angle is less than 180 degrees. It can be represented as a matrix, where each entry in the matrix corresponds to a square in the polyomino.

Q: What is a sub-square?

A: A sub-square is a square region within the diagram, where each side is parallel to the sides of the diagram. The size of a sub-square is determined by the number of squares it contains.

Q: How do you count the number of sub-squares in a Ferrers diagram?

A: To count the number of sub-squares in a Ferrers diagram, you can use a recursive formula. The formula is:

C(n,h)=∑i=0n−hC(i,h)⋅C(n−i,h)C(n, h) = \sum_{i=0}^{n-h} C(i, h) \cdot C(n-i, h)

where $ C(n, h) $ is the number of sub-squares of size $ h \times h $ in a Ferrers diagram of size $ n \times n $.

Q: How do you count the number of sub-squares in a convex polyomino?

A: To count the number of sub-squares in a convex polyomino, you can use a similar recursive formula. The formula is:

P(n,h)=∑i=0n−hP(i,h)⋅P(n−i,h)P(n, h) = \sum_{i=0}^{n-h} P(i, h) \cdot P(n-i, h)

where $ P(n, h) $ is the number of sub-squares of size $ h \times h $ in a convex polyomino of size $ n \times n $.

Q: What is the asymptotic behavior of the number of sub-squares?

A: The asymptotic behavior of the number of sub-squares is given by the following theorems:

Theorem 1: Let $ C(n, h) $ be the number of sub-squares of size $ h \times h $ in a Ferrers diagram of size $ n \times n $. Then, for fixed $ h $, we have:

C(n,h)∼1h2⋅n2C(n, h) \sim \frac{1}{h^2} \cdot n^2

as $ n \to \infty $.

Theorem 2: Let $ P(n, h) $ be the number of sub-squares of size $ h \times h $ in a convex polyomino of size $ n \times n $. Then, for fixed $ h $, we have:

P(n,h)∼1h2⋅n2P(n, h) \sim \frac{1}{h^2} \cdot n^2

as $ n \to \infty $.

Q: What are some applications of counting sub-squares?

A: Counting sub-squares has applications in various fields, including:

  • Computer science: Counting sub-squares can be used to study the properties algorithms and data structures.
  • Mathematics: Counting sub-squares can be used to study the properties of combinatorial objects, such as Ferrers diagrams and convex polyominoes.
  • Physics: Counting sub-squares can be used to study the properties of physical systems, such as lattice gases and spin systems.

Q: What are some open problems in counting sub-squares?

A: Some open problems in counting sub-squares include:

  • Studying the properties of sub-squares in higher-dimensional spaces.
  • Analyzing the asymptotic behavior of the number of sub-squares in more general settings.
  • Developing new algorithms for counting sub-squares.

Q: Where can I learn more about counting sub-squares?

A: You can learn more about counting sub-squares by reading the following resources:

  • [1] Stanley, R. P. (1999). Enumerative Combinatorics, Vol. 2. Cambridge University Press.
  • [2] Klarner, D. A. (1971). Polyominoes and their enumeration. Journal of Combinatorial Theory, Series A, 11(2), 147-164.
  • [3] Viennot, G. (1983). Une formule pour le nombre de polyominos convexes. Comptes Rendus de l'Académie des Sciences, Série A, 296(12), 531-534.