Let hn denote the number of ways to color the squares of a 1...

Let $h_{n}$ denote the number of ways to color the squares of a 1 -by- $n$ board with the colors red, white, blue, and green in such a way that the number of squares colored red is even and the number of squares colored white is odd. Determine the exponential generating function for the sequence $h_{0}, h_{1}, \ldots, h_{n}, \ldots$, and then find a simple formula for $h_{n}$.

Key Concepts

Exponential Generating Function

An exponential generating function (EGF) is a power series used in combinatorics where the nth term of a sequence is encoded as a coefficient multiplied by x^n?n!. EGFs are particularly useful when dealing with problems where the order of elements matters or when dealing with labeled structures, as they facilitate the translation of combinatorial constructions into algebraic manipulations that can account for additional weights or conditions.

Parity Constraints

Parity constraints refer to conditions in a counting problem that require certain counts to be even or odd. In many combinatorial problems, these restrictions are imposed to narrow the set of allowable configurations. By incorporating parity conditions into generating functions, one can separately track contributions from even and odd counts, often by introducing alternating signs or distinguishing between cases, which aids in extracting the final count that satisfies the specified constraints.

Combinatorial Coloring Problems

Combinatorial coloring problems involve counting the number of ways to assign colors or labels to objects (such as the squares of a board) subject to certain restrictions. These problems can combine basic counting principles with additional conditions, such as specific counts for each color or restrictions on adjacent colors. Generating functions are a powerful tool in these problems because they allow the combination of different choices and constraints into a unified algebraic expression.

Derivation of Closed-Form Expressions

In combinatorial enumeration, deriving a closed-form expression for a sequence means finding a simple, explicit formula for the nth term, rather than a recursive or implicit definition. This often involves manipulating generating functions—like the EGF—to extract coefficients in a form that can be simplified into an analytic expression. Achieving a closed-form solution not only gives direct insight into the behavior of the sequence but also simplifies computations for larger values of n.

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