A326763 Number of permutations of length n and order at most 3 whose powers all avoid the pattern 132.

1, 1, 2, 5, 10, 16, 36, 65, 118, 232, 452, 800, 1622, 3042, 5758, 11077, 21712, 40204, 79718, 151628, 292994, 561954, 1103786, 2087696, 4115506, 7884446, 15393710, 29592074, 58229334, 111422134, 219575234, 422888473, 830617400, 1602832900, 3160618558, 6092881976

Offset

0,3

Links

Table of n, a(n) for n=0..35.

Miklós Bóna and Rebecca Smith, Pattern Avoidance in Permutations and Their Squares, Discrete Mathematics, 342 (2019), pp. 3194-3200; arXiv:1901.00026 [math.CO], 2019.

Amanda Burcroff and Colin Defant, Pattern-Avoiding Permutation Powers, Discrete Mathematics, 343 (2020), 112017; arXiv:1907.09451 [math.CO], 2019.

Formula

a(n) = A014495(n) + A370686(n), where the 1st (resp. 2nd) term counts 132-avoiding permutations of order 2 (resp. 1 or 3). - Andrey Zabolotskiy, Apr 13 2025

Prog

(SageMath)
def a(n):
    return len([p for p in Permutations(n)
        if p*p == Permutations(n).identity() and p.avoids([1, 3, 2])
        or p*p*p == Permutations(n).identity() and p.avoids([1, 3, 2]) and (p*p).avoids([1, 3, 2])]) # Andrey Zabolotskiy, Apr 13 2025

Crossrefs

Cf. A326760, A326761, A326762, A001405, A014495, A370686.

Sequence in context: A192701 A258601 A264300 * A067112 A101306 A051351

Adjacent sequences: A326760 A326761 A326762 * A326764 A326765 A326766

Keyword

nonn

Author

Amanda Burcroff, Aug 15 2019

Extensions

Terms a(16) onwards using the formula from Andrey Zabolotskiy, Apr 14 2025

Status

approved