Pairing with a difference of 1 or 11

HardCombinatorics Brainteasers~12m

Let $S = \{1, 2, 3, \dots, 22\}$ . In how many ways can $S$ be partitioned into $11$ pairs such that the two elements of every pair differ by exactly $1$ or exactly $11$ ?

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Count perfect matchings of the graph on $1$ – $22$ with edges for differences $1$ and $11$ .
Recurse: the smallest unmatched vertex pairs with $+1$ or $+11$ ; sum the branches to get $145$ .

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145

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Pairing with a difference of 1 or 11

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