cp's OEIS Frontend

A348379 Number of factorizations of n with an alternating permutation.

%I A348379 #12 Nov 03 2021 10:48:45
%S A348379 1,1,1,1,1,2,1,2,1,2,1,4,1,2,2,3,1,4,1,4,2,2,1,6,1,2,2,4,1,5,1,5,2,2,
%T A348379 2,8,1,2,2,6,1,5,1,4,4,2,1,10,1,4,2,4,1,6,2,6,2,2,1,11,1,2,4,6,2,5,1,
%U A348379 4,2,5,1,15,1,2,4,4,2,5,1,10,3,2,1,11,2
%N A348379 Number of factorizations of n with an alternating permutation.
%C A348379 First differs from A335434 at a(216) = 27, A335434(216) = 28. Also differs from A335434 at a(270) = 19, A335434(270) = 20.
%C A348379 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A348379 All of the counted factorizations are separable (A335434).
%C A348379 A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.
%H A348379 Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>
%F A348379 a(2^n) = A345170(n).
%e A348379 The a(270) = 19 factorizations:
%e A348379   (2*3*3*15)  (2*3*45)  (2*135)  (270)
%e A348379   (2*3*5*9)   (2*5*27)  (3*90)
%e A348379   (3*3*5*6)   (2*9*15)  (5*54)
%e A348379               (3*3*30)  (6*45)
%e A348379               (3*5*18)  (9*30)
%e A348379               (3*6*15)  (10*27)
%e A348379               (3*9*10)  (15*18)
%e A348379               (5*6*9)
%t A348379 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A348379 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A348379 Table[Length[Select[facs[n],Select[Permutations[#],wigQ]!={}&]],{n,100}]
%Y A348379 Partitions not of this type are counted by A345165, ranked by A345171.
%Y A348379 Partitions of this type are counted by A345170, ranked by A345172.
%Y A348379 Twins and partitions of this type are counted by A344740, ranked by A344742.
%Y A348379 The case with twins is A347050.
%Y A348379 The complement is counted by A348380, without twins A347706.
%Y A348379 The ordered version is A348610.
%Y A348379 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A348379 A001250 counts alternating permutations.
%Y A348379 A025047 counts alternating or wiggly compositions, ranked by A345167.
%Y A348379 A339846 counts even-length factorizations.
%Y A348379 A339890 counts odd-length factorizations.
%Y A348379 Cf. A038548, A056986, A325534, A335452, A347437, A347438, A347439, A347442, A347456, A347463, A348381, A348383, A348609.
%K A348379 nonn
%O A348379 1,6
%A A348379 _Gus Wiseman_, Oct 28 2021