This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348610 #6 Nov 06 2021 02:26:25
%S A348610 1,1,1,1,1,3,1,3,1,3,1,6,1,3,3,4,1,6,1,6,3,3,1,12,1,3,3,6,1,11,1,7,3,
%T A348610 3,3,15,1,3,3,12,1,11,1,6,6,3,1,23,1,6,3,6,1,12,3,12,3,3,1,28,1,3,6,
%U A348610 12,3,11,1,6,3,11,1,33,1,3,6,6,3,11,1,23,4,3
%N A348610 Number of alternating ordered factorizations of n.
%C A348610 An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
%C A348610 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).
%H A348610 Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>
%e A348610 The alternating ordered factorizations of n = 1, 6, 12, 16, 24, 30, 32, 36:
%e A348610 () 6 12 16 24 30 32 36
%e A348610 2*3 2*6 2*8 3*8 5*6 4*8 4*9
%e A348610 3*2 3*4 8*2 4*6 6*5 8*4 9*4
%e A348610 4*3 2*4*2 6*4 10*3 16*2 12*3
%e A348610 6*2 8*3 15*2 2*16 18*2
%e A348610 2*3*2 12*2 2*15 2*8*2 2*18
%e A348610 2*12 3*10 4*2*4 3*12
%e A348610 2*4*3 2*5*3 2*6*3
%e A348610 2*6*2 3*2*5 2*9*2
%e A348610 3*2*4 3*5*2 3*2*6
%e A348610 3*4*2 5*2*3 3*4*3
%e A348610 4*2*3 3*6*2
%e A348610 6*2*3
%e A348610 2*3*2*3
%e A348610 3*2*3*2
%t A348610 ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
%t A348610 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]] == Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A348610 Table[Length[Select[ordfacs[n],wigQ]],{n,100}]
%Y A348610 The additive version (compositions) is A025047 ranked by A345167.
%Y A348610 The complementary additive version is A345192, ranked by A345168.
%Y A348610 Dominated by A348611 (the anti-run version) at positions A122181.
%Y A348610 The complement is counted by A348613.
%Y A348610 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A348610 A001250 counts alternating permutations, complement A348615.
%Y A348610 A339846 counts even-length factorizations.
%Y A348610 A339890 counts odd-length factorizations.
%Y A348610 A345165 counts partitions w/o an alternating permutation, ranked by A345171.
%Y A348610 A345170 counts partitions w/ an alternating permutation, ranked by A345172.
%Y A348610 A347463 counts ordered factorizations with integer alternating product.
%Y A348610 A348379 counts factorizations w/ an alternating permutation.
%Y A348610 A348380 counts factorizations w/o an alternating permutation.
%Y A348610 Cf. A038548, A049774, A056986, A102726, A128761, A344614, A347050, A347464, A347706, A348383.
%K A348610 nonn
%O A348610 1,6
%A A348610 _Gus Wiseman_, Nov 05 2021