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

0,3

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.

Links

Table of n, a(n) for n=0..26.

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

Example

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

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).

Sequence in context: A340638 A177905 A118405 * A095912 A143474 A261341

Adjacent sequences: A332342 A332343 A332344 * A332346 A332347 A332348

Keyword

nonn

Author

David Bevan, Feb 10 2020

Status

approved