A348615 Number of non-alternating permutations of {1...n}.

0, 0, 0, 2, 14, 88, 598, 4496, 37550, 347008, 3527758, 39209216, 473596070, 6182284288, 86779569238, 1303866853376, 20884006863710, 355267697410048, 6397563946377118, 121586922638606336, 2432161265800164950, 51081039175603191808, 1123862030028821404198

Offset

0,4

Comments

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.

Also permutations of {1...n} matching the consecutive patterns (1,2,3) or (3,2,1). Matching only one of these gives A065429.

Links

Table of n, a(n) for n=0..22.

Wikipedia, Alternating permutation

Formula

a(n) = n! - A001250(n).

Example

The a(4) = 14 permutations:
  (1,2,3,4)  (3,1,2,4)
  (1,2,4,3)  (3,2,1,4)
  (1,3,4,2)  (3,4,2,1)
  (1,4,3,2)  (4,1,2,3)
  (2,1,3,4)  (4,2,1,3)
  (2,3,4,1)  (4,3,1,2)
  (2,4,3,1)  (4,3,2,1)

Maple

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> n!-`if`(n<2, 1, 2)*b(n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 04 2021

Mathematica

wigQ[y_]:=Or[Length[y]==0, Length[Split[y]] ==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Permutations[Range[n]], !wigQ[#]&]], {n, 0, 6}]

Prog

(Python)
from itertools import accumulate, count, islice
def A348615_gen(): # generator of terms
    yield from (0, 0)
    blist, f = (0, 2), 1
    for n in count(2):
        f *= n
        yield f - (blist := tuple(accumulate(reversed(blist), initial=0)))[-1]
A348615_list = list(islice(A348615_gen(), 40)) # Chai Wah Wu, Jun 09-11 2022

Crossrefs

The complement is counted by A001250, ranked by A333218.

The complementary version for compositions is A025047, ranked by A345167.

A directed version is A065429, complement A049774.

The version for compositions is A345192, ranked by A345168.

The version for ordered factorizations is A348613, complement A348610.

A345165 counts partitions w/o an alternating permutation, ranked by A345171.

A345170 counts partitions w/ an alternating permutation, ranked by A345172.

A348379 counts factorizations with an alternating permutation.

A348380 counts factorizations without an alternating permutation.

Cf. A056986, A102726, A325534, A325535, A344614, A344653, A344654, A347050, A347706, A348377, A348609.

Sequence in context: A005610 A065355 A162478 * A189392 A235374 A065892

Adjacent sequences: A348612 A348613 A348614 * A348616 A348617 A348618

Keyword

nonn

Author

Gus Wiseman, Nov 03 2021

Status

approved