A141110 Number of cycles and fixed points in the permutation (n, n-2, n-4, ..., 1, ..., n-3, n-1).

1, 1, 1, 2, 1, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 4, 3, 1, 3, 2, 3, 5, 1, 2, 5, 1, 3, 4, 1, 1, 7, 6, 1, 3, 1, 4, 5, 3, 1, 4, 1, 7, 3, 4, 5, 7, 3, 2, 7, 1, 1, 8, 1, 3, 3, 4, 3, 7, 5, 2, 5, 3, 9, 10, 1, 5, 7, 2, 1, 3, 3, 6, 5, 1, 5, 8, 7, 3, 3, 4, 1, 9, 1, 2, 11

Offset

1,4

Comments

The above permutation (see A130517) can be generated by taking S_n: (1, 2, ..., n) and reversing the first two, first three and so on till first n, elements in sequence. Interestingly this permutation orbit has length given by A003558.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Example

a(20) = 2, since (20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19) has two cycles (1, 20, 19, 17, 13, 5, 12, 3, 16, 11) and (2, 18, 15, 9, 4, 14, 7, 8, 6, 10).

Prog

(Python)
from sympy.combinatorics import Permutation
def a(n):
    p = list(range(n, 0, -2)) + list(range(1+(n%2), n, 2))
    return Permutation([pi-1 for pi in p]).cycles
print([a(n) for n in range(1, 86)]) # Michael S. Branicky, Dec 27 2021

Crossrefs

Cf. A130517 (permutations), A003558 (order).

Sequence in context: A329359 A235682 A324830 * A325758 A280073 A327523

Adjacent sequences: A141107 A141108 A141109 * A141111 A141112 A141113

Keyword

easy,nonn

Author

Ramasamy Chandramouli, Jun 05 2008

Status

approved