A348615 - OEIS

%I #17 Jun 11 2022 11:43:08

%S 0,0,0,2,14,88,598,4496,37550,347008,3527758,39209216,473596070,

%T 6182284288,86779569238,1303866853376,20884006863710,355267697410048,

%U 6397563946377118,121586922638606336,2432161265800164950,51081039175603191808,1123862030028821404198

%N Number of non-alternating permutations of {1...n}.

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

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>

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

%e The a(4) = 14 permutations:

%e   (1,2,3,4)  (3,1,2,4)

%e   (1,2,4,3)  (3,2,1,4)

%e   (1,3,4,2)  (3,4,2,1)

%e   (1,4,3,2)  (4,1,2,3)

%e   (2,1,3,4)  (4,2,1,3)

%e   (2,3,4,1)  (4,3,1,2)

%e   (2,4,3,1)  (4,3,2,1)

%p b:= proc(u, o) option remember;

%p       `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))

%p     end:

%p a:= n-> n!-`if`(n<2, 1, 2)*b(n, 0):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 04 2021

%t wigQ[y_]:=Or[Length[y]==0,Length[Split[y]] ==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];

%t Table[Length[Select[Permutations[Range[n]],!wigQ[#]&]],{n,0,6}]

%o (Python)

%o from itertools import accumulate, count, islice

%o def A348615_gen(): # generator of terms

%o     yield from (0,0)

%o     blist, f = (0,2), 1

%o     for n in count(2):

%o         f *= n

%o         yield f - (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]

%o A348615_list = list(islice(A348615_gen(),40)) # _Chai Wah Wu_, Jun 09-11 2022

%Y The complement is counted by A001250, ranked by A333218.

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

%Y A directed version is A065429, complement A049774.

%Y The version for compositions is A345192, ranked by A345168.

%Y The version for ordered factorizations is A348613, complement A348610.

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

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

%Y A348379 counts factorizations with an alternating permutation.

%Y A348380 counts factorizations without an alternating permutation.

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Nov 03 2021