We enumerate and characterize some classes of alternating reverse involutions avoiding a single pattern length three or four. If on one hand the case patterns is trivial, other hand, four more challenging involves sequences combinatorial interest, such as Motzkin Fibonacci numbers.

In this paper, we investigate the existence of free involutions on some Wall manifolds and compute mod 2 cohomology algebra correspondent orbit space.

We prove Borchardt’s identity det ( 1 xi − yj ) per ( 1 xi − yj ) = det ( 1 (xi − yj) ) by means of sign-reversing involutions.