A309504
Number of cyclic permutations of length n avoiding the pattern 123.
Original entry on oeis.org
1, 1, 1, 2, 4, 10, 24, 68, 188, 586, 1722, 5492, 16924, 55582, 177278, 594460, 1944980, 6628384, 22132112, 76421498, 259359036, 905416294, 3114033930, 10971347070, 38157201530
Offset: 0
For n=3, there are two such permutations, 231 and 312.
The a(4) = 4 permutations are: 2413, 3142, 3421, 4312.
The a(5) = 10 permutations are: 25413, 35214, 35421, 41532, 43152, 43521, 45231, 53412, 54132, 54213.
- Kassie Archer, Christina Graves, and Robert Laudone, Binary operations on pattern-avoiding cycles, arXiv:2505.04456 [math.CO], 2025. See p. 23.
- Miklos Bona and Michael Cory, Cyclic Permutations Avoiding Pairs of Patterns of Length Three, arXiv:1805.05196 [math.CO], 2018.
- Andrew Howroyd, PARI Program, Nov 2024.
Cf.
A000108 (number of permutations avoiding 123).
-
\\ See Links for program code.
for(n=0, 16, print1(E123(n), ", ")) \\ Andrew Howroyd, Nov 20 2024
A309505
Number of cyclic permutations of length n that avoid the pattern 132 (equivalently, 213).
Original entry on oeis.org
1, 1, 1, 2, 4, 10, 24, 68, 182, 544, 1574, 4888, 14864, 47610, 149964, 491802, 1592198, 5318936, 17593170, 59679516, 200805614, 689988886, 2354489616, 8178944510, 28240716098
Offset: 0
For n=3, there are two such permutations, 231 and 312.
The a(4) = 4 permutations are: 2341, 3421, 4123, 4312.
The a(5) = 10 permutations are: 23451, 34251, 34512, 43521, 45123, 45231, 51234, 53124, 53412, 54213.
- 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.
- Brice Huang, An Upper Bound on the Number of (132,213)-Avoiding Cyclic Permutations, arXiv:1808.08462 [math.CO], 2018-2019.
Cf.
A000108 (number of permutations avoiding 132).
-
\\ See PARI link in A309504 for program code.
for(n=1, 16, print1(E213(n), ", ")) \\ Andrew Howroyd, Nov 20 2024
A309506
Number of cyclic permutations of length n avoiding the pattern 231 (equivalently, 312).
Original entry on oeis.org
1, 1, 1, 1, 2, 5, 12, 30, 86, 253, 748, 2274, 7152, 22890, 74189, 243342, 808599, 2716549, 9213420, 31498358, 108483093, 376145636, 1312463081, 4605569378, 16245866825
Offset: 0
For n=4, there are two such permutations, 4123 and 4312.
The a(5) = 5 permutations are 51234, 51423, 53124, 54132, 54213.
Cf.
A000108 (number of permutations avoiding 231).
-
\\ See PARI link in A309504 for program code.
for(n=0, 16, print1(E231(n), ", ")) \\ Andrew Howroyd, Nov 20 2024
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