On the number of $\tau $-tilting modules over the Auslander algebras of radical square zero Nakayama algebras

Let $\Lambda _n$ be a radical square zero Nakayama algebra with $n$ simple modules and $\Gamma _n$ the Auslander algebra of $\Lambda _n$. We calculate the number $|\tau \text {-tilt}\,\Gamma _n|$ of $\tau $-tilting modules and the number $|{\rm s}\tau \text {-tilt}\,\Gamma _n|$ of support $\tau $-tilting modules over $\Gamma _n$. In particular, we prove the recurrence relations $$|\tau \text {-tilt}\,\Gamma _n|=3|\tau \text {-tilt}\,\Gamma _{n-1}|+|\tau \text {-tilt}\,\Gamma _{n-2}|,$$ $$|{\rm s}\tau \text {-tilt}\,\Gamma _n|=6|{\rm s}\tau \text {-tilt}\,\Gamma _{n-1}|+3|{\rm s}\tau \text {-tilt}\,\Gamma _{n-2}|,$$ from which the exact values of $|\tau \text {-tilt}\,\Gamma _n|$ and $|{\rm s}\tau \text {-tilt}\,\Gamma _n|$ are derived.