A295264 Number of total cyclic orders Z on {0, ..., n-1} such that (i, (i+1) mod n, (i+2) mod n) in Z for 0 <= i < n.

1, 1, 1, 1, 2, 9, 31, 128, 708, 4015, 24865, 177444, 1357830, 11141634, 99680595, 953369248, 9687797896, 104909705019, 1202985610821, 14542305462860, 185165060805578, 2476043157780478, 34673872424313463

Offset: 1

Author

Eric M. Schmidt, Nov 19 2017

Keywords

• nonn

Comments

For all n >= 1, a(n) is the number of n-sided polygons, turning always in the same direction (right or left) while following its edges. By "n-sided polygons" we mean the polygons that can be drawn by connecting n equally spaced points on a circle. - Ludovic Schwob, Apr 04 2021

For all n >= 1, a(n) is the number of cyclic permutations of length n that avoid consecutive patterns 123, 231, and 312. - Rupert Li, Jul 27 2021

Links

• Ludovic Schwob, Table of n, a(n) for n = 1..39
• Rupert Li, Vincular Pattern Avoidance on Cyclic Permutations, arXiv:2107.12353 [math.CO], 2021.
• Sanjay Ramassamy, Extensions of partial cyclic orders, Euler numbers and multidimensional boustrophedons, arXiv:1706.03386 [math.CO], 2017.
• Ludovic Schwob, Illustration of a(7) and a(8)

Crossrefs

Row sums of A343257.

Programs

PARI

\\ Needs B function from A343257.
a(n)={sum(i=1, n, B(n,i,1))} \\ Andrew Howroyd, May 16 2021

Extensions

a(12) corrected and a(13)-a(18) from Andrew Howroyd, May 15 2021

Corrected initial offset/terms by Rupert Li, Sep 17 2021

a(19) onwards from Ludovic Schwob, Oct 31 2022