A309508 Number of cyclic permutations of length n avoiding the pattern 321.

1, 1, 1, 2, 4, 10, 24, 66, 178, 512, 1486, 4446, 13468, 41648, 130178, 412670, 1321418, 4274970, 13948966, 45890440, 152061154, 507292698, 1702753462, 5748085332, 19506240462

Offset: 0

Author

Miklos Bona, Aug 05 2019

Keywords

• nonn
• more

Comments

Comment from F. Chapoton, Sep 14 2021: (Start)

The maps sending a permutation to its inverse or to its reverse-complement define two commuting involutions on these sets of permutations.

The next terms in the sequence could be 41648, 130178, though these are counting Dyck words such that an associated permutation is cyclic, related but not obviously equivalent combinatorial objects. (End)

Examples

			For n=3, there are two such permutations, 231 and 312.
The a(4) = 4 permutations are: 2341, 2413, 3142, 4123.
The a(5) = 10 permutations are: 23451, 23514, 24153, 25134, 31452, 31524, 34512, 41253, 45123, 51234.
		

Links

• Kassie Archer, Christina Graves, and Robert Laudone, Binary operations on pattern-avoiding cycles, arXiv:2505.04456 [math.CO], 2025.
• Miklos Bona and Michael Cory, Cyclic Permutations Avoiding Pairs of Patterns of Length Three, arXiv:1805.05196 [math.CO], 2018.

Crossrefs

Cf. A000108 (number of permutations avoiding 321).

Cf. A000957, A309504, A309505, A309506.

Programs

PARI

\\ See PARI link in A309504 for program code.
for(n=0, 16, print1(E321(n), ", ")) \\ Andrew Howroyd, Nov 20 2024

Extensions

a(0)=1 prepended and a(13)-a(24) from Andrew Howroyd, Nov 17 2024