A332345 a(n) is the number of totally alternating permutations of 1,2,...,n.

1, 1, 2, 4, 6, 12, 28, 76, 240, 852, 3392, 14868, 71392, 371740, 2089184, 12589980, 81037792, 554555300, 4021728992, 30800978980, 248464000480, 2105271605804, 18696216008416, 173630895701996, 1683187452989920, 17000599430003444, 178625854452674272

Offset: 0

Author

David Bevan, Feb 10 2020

Keywords

• nonn

Comments

A permutation w of 1,...,n is totally alternating if both w and w^{-1} are either alternating or reverse alternating.

Totally alternating permutations are those that avoid the consecutive patterns 123 and 321 and also the consecutive covincular patterns 123 and 321.

If a permutation is totally alternating, then so are its other 7 symmetries.

Examples

			The six totally alternating permutations of 1,...,4 are 1324, 2143, 2413, 3142, 3412 and 4231.
		

Links

• R. P. Stanley, Alternating permutations and symmetric functions, arXiv:math/0603520 [math.CO], 2006.

Crossrefs

For n > 1, a(n) = 2*(A007999(n) + A332344(n)).

For odd n > 1, a(n) = 4*A007999(n).

For even n > 1, a(n) = 4*A007999(n) - 2*A007999(n-2).