Combinatorics And Applications Problem Set 31 Let An Be The

Combinatorics And Applicationsproblem Set 31 Let An Be The Number Of

Consider the collection of problems related to generating functions, recurrence relations, and combinatorial counting. The tasks involve finding recurrence relations for specific counting problems, deriving closed-form expressions for generating functions as ratios of polynomials, and applying these methods to classic combinatorial scenarios such as tiling problems and making change with coins.

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These problems exemplify fundamental techniques in combinatorics, including the derivation and manipulation of generating functions and solving recurrence relations. They demonstrate how to model counting problems using algebraic tools and how to interpret the combinatorial meaning of generating functions. The solutions to these problems can provide insight into the enumeration of arrangements, partitions, and resource distributions, which are essential in discrete mathematics, computer science, and related fields.

Problem 1: Tiling a 1×n Board with Tiles of Sizes 1×1, 1×2, and 1×3

Let an denote the number of ways to tile a 1×n board with tiles of sizes 1×1, 1×2, and 1×3. To find the recurrence relation, consider how to construct a tiling of length n:

• If the first tile is a 1×1, then the remaining tiles form a tiling of length n−1.
• If the first tile is a 1×2, then the remaining tiles form a tiling of length n−2.
• If the first tile is a 1×3, then the remaining tiles form a tiling of length n−3.

Hence, the recurrence relation is:

an = an−1 + an−2 + an−3

with initial conditions based on small boards:

• a0 = 1 (empty board has exactly one way to tile it)
• a1 = 1 (only one way to tile a 1×1 board with a 1×1 tile)
• a2 = 2 (two ways: two 1×1 tiles or one 1×2 tile)

To find the generating function G(z) = ∑_n≥0 a_n zⁿ, use the recurrence relation:

G(z) = a0 + a1 z + a2 z² + ∑_n≥3 a_n zⁿ

Substituting the recurrence:

G(z) - a0 - a1 z - a2 z² = z (G(z) - a0) + z² G(z) + z³ G(z)

Simplify and solve for G(z):

G(z) (1 - z - z² - z³) = 1 + z

Therefore:

G(z) = (1 + z) / (1 - z - z² - z³)

Problem 2: Recurrence with Initial Conditions

Given the recurrence relation an = 3a_n−1 - 4a_n−3, with initial conditions a0=1, a1=2, a2=6:

To find a closed-form generating function, define G(z) = ∑_n≥0 a_n zⁿ. Using the recurrence, multiply both sides by zⁿ and sum over n:

∑_n≥0 a_n zⁿ = ∑_n≥0 (3a_n−1 - 4a_n−3) zⁿ

Adjust the sums for initial terms and shifts:

G(z) = 3z G(z) + 4z³ G(z) + polynomial terms from initial conditions

Expressed explicitly, the generating function becomes:

G(z) = (a0 + (a1 - 3a0)z + (a2 - 3a1 + 4a0)z²) / (1 - 3z + 4z³)

Substituting initial values:

G(z) = (1 + (2 - 31)z + (6 - 32 + 4*1)z²) / (1 - 3z + 4z³)

which simplifies to:

G(z) = (1 - z + 6z²) / (1 - 3z + 4z³)

Problem 3: Explicit Formula for a_n

Given a_n = 5·3ⁿ − 2ⁿ⁺¹, for n ≥ 0:

The generating function G(z) = ∑_n≥0 a_n zⁿ separates into two basic geometric series:

G(z) = 5 ∑_n≥0 (3ⁿ) zⁿ - 2 ∑_n≥0 (2ⁿ⁺¹) zⁿ

Express each sum explicitly:

G(z) = 5 / (1 - 3z) - 2 ∑_n≥0 2 · (2ⁿ) zⁿ = 5 / (1 - 3z) - 4 / (1 - 2z)

Problem 4: Number of Ways to Make Change for a Dollar

Calculate the generating function A(z) representing the number of ways to give change of 100 cents using coins of denominations 1, 5, 10, and 25 cents. Each coin's generating function is:

• 1-cent coins: 1 + z + z² + ... = 1 / (1 - z)
• 5-cent coins: 1 + z⁵ + z¹⁰ + ... = 1 / (1 - z⁵)
• 10-cent coins: 1 + z¹⁰ + ... = 1 / (1 - z¹⁰)
• 25-cent coins: 1 + z²⁵ + ... = 1 / (1 - z²⁵)

The combined generating function is:

A(z) = 1 / [(1 - z)(1 - z⁵)(1 - z¹⁰)(1 - z²⁵)]

The coefficient of z¹⁰⁰ in this expansion gives the number of ways to make change for one dollar.

Problem 5: Generating Function for n·zⁿ

Find the closed-form expression for the generating function S(z) = ∑_n=1^∞ n zⁿ. Recognize that this is a standard power series:

S(z) = z / (1 - z)²

This arises from differentiation of the geometric series and captures the sum involving the linear term n.

Conclusion

These problems demonstrate the versatility and power of generating functions in combinatorial analysis. The explicit formulas and recurrence relations serve as fundamental tools for counting arrangements, tiling configurations, and resource allocation scenarios in discrete mathematics. A firm grasp of these techniques enhances problem-solving capabilities across theoretical and applied domains in computer science, mathematics, and related disciplines.

References

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