# Number of Standard Young Tableaux of n cells

I know there is a 1-1 correspondance between the number of standard young tableaux of $n$ cells and the number of involutions in $S_n$. Number of involutions in $S_n$ satisfies the recurrence relation $$a_{n+1}=a_n+na_{n-1}$$ How can we prove that the number of standard young tableaux of $n$ cells also satisfies this relation without using the correspondance with the number of involutions?

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