MathDB

For any integer

n\geq1

, we consider a set

P_{2n}

2n

points placed equidistantly on a circle. A perfect matching on this point set is comprised of

n

(straight-line) segments whose endpoints constitute

P_{2n}

. Let

\mathcal{M}_{n}

denote the set of all non-crossing perfect matchings on

P_{2n}

. A perfect matching

M\in \mathcal{M}_{n}

is said to be centrally symmetric, if it is invariant under point reflection at the circle center. Determine, as a function of

n

, the number of centrally symmetric perfect matchings within

\mathcal{M}_{n}

.
Proposed by Mirko Petrusevski

Problem Statement