Call a subset S of the integers {1,2, …, n} extraordinary provided its smallest integer equals i

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Call a subset S of the integers {1,2, …, n} extraordinary...

Call a subset $S$ of the integers $\{1,2, \ldots, n\}$ extraordinary provided its smallest integer equals its size: $$\min \{x: x \in S\}=|S|$$ For example, $S=\{3,7,8\}$ is extraordinary. Let $g_{n}$ be the number of extraordinary subsets of $\{1,2, \ldots, n\}$. Prove that $$g_{n}=g_{n-1}+g_{n-2}, \quad(n \geq 3)$$ with $g_{1}=1$ and $g_{2}=1$.

Call a subset $S$ of the integers $\{1,2, \ldots, n\}$ extraordinary provided its smallest integer equals its size:
$$\min \{x: x \in S\}=|S|$$
For example, $S=\{3,7,8\}$ is extraordinary. Let $g_{n}$ be the number of extraordinary subsets of $\{1,2, \ldots, n\}$. Prove that
$$g_{n}=g_{n-1}+g_{n-2}, \quad(n \geq 3)$$
with $g_{1}=1$ and $g_{2}=1$.

Key Concepts

Recurrence Relations

A recurrence relation defines each term of a sequence as a function of one or more previous terms. In this problem, the extraordinary subsets are counted recursively by relating the count for n elements to the counts for n-1 and n-2 elements, mirroring the structure of many standard recursive sequences such as the Fibonacci numbers.

Combinatorial Case Analysis

Case analysis in combinatorics involves breaking down a counting problem into distinct, non-overlapping cases whose counts can be calculated separately. In this context, the division of extraordinary subsets based on the presence or absence of certain elements allows the problem to be split into simpler parts that relate directly to the recurrence.

Subset Selection with Constraints

Counting subsets with additional constraints is a fundamental concept in combinatorics. Here, the constraint is that the smallest element of the subset must equal its size. This restriction uniquely defines the structure of the subsets being counted and is key to establishing the recurrence relation.

Combinatorial Proof Techniques

Combinatorial proofs use counting arguments to establish identities or relationships, such as recurrence relations. They often rely on constructing bijections or partitioning sets in a clever way to match the structure of a recursive relationship, as seen in this problem where split cases are directly related to g_{n-1} and g_{n-2}.

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