cp's OEIS Frontend

A348609 Numbers with a separable factorization without an alternating permutation.

%I A348609 #11 Nov 02 2021 21:00:15
%S A348609 216,270,324,378,432,486,540,594,640,648,702,756,768,810,864,896,918,
%T A348609 960,972,1024,1026,1080,1134,1152,1188,1242,1280,1296,1344,1350,1404,
%U A348609 1408,1458,1500,1512,1536,1566,1620,1664,1674,1728,1750,1782,1792,1836,1890
%N A348609 Numbers with a separable factorization without an alternating permutation.
%C A348609 A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
%C A348609 A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts. Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of the remaining multiplicities plus one.
%C A348609 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 sets.
%C A348609 Note that 216 has separable prime factorization (2*2*2*3*3*3) with an alternating permutation, but the separable factorization (2*3*3*3*4) is has no alternating permutation. See also A345173.
%e A348609 The terms and their prime factorizations begin:
%e A348609   216 = 2*2*2*3*3*3
%e A348609   270 = 2*3*3*3*5
%e A348609   324 = 2*2*3*3*3*3
%e A348609   378 = 2*3*3*3*7
%e A348609   432 = 2*2*2*2*3*3*3
%e A348609   486 = 2*3*3*3*3*3
%e A348609   540 = 2*2*3*3*3*5
%e A348609   594 = 2*3*3*3*11
%e A348609   640 = 2*2*2*2*2*2*2*5
%e A348609   648 = 2*2*2*3*3*3*3
%e A348609   702 = 2*3*3*3*13
%e A348609   756 = 2*2*3*3*3*7
%e A348609   768 = 2*2*2*2*2*2*2*2*3
%e A348609   810 = 2*3*3*3*3*5
%e A348609   864 = 2*2*2*2*2*3*3*3
%t A348609 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A348609 sepQ[m_]:=Select[Permutations[m],!MatchQ[#,{___,x_,x_,___}]&]!={};
%t A348609 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A348609 Select[Range[1000],Function[n,Select[facs[n],sepQ[#]&&Select[Permutations[#],wigQ]=={}&]!={}]]
%Y A348609 Partitions of this type are counted by A345166, ranked by A345173 (a superset).
%Y A348609 Compositions of this type are counted by A345195, ranked by A345169.
%Y A348609 A001055 counts factorizations, strict A045778, ordered A074206.
%Y A348609 A001250 counts alternating permutations, complement A348615.
%Y A348609 A025047 counts alternating compositions, complement A345192, ranked by A345167.
%Y A348609 A335434 counts separable factorizations, with twins A348383, complement A333487.
%Y A348609 A339846 counts even-length factorizations.
%Y A348609 A339890 counts odd-length factorizations.
%Y A348609 A345165 counts partitions w/o an alternating permutation, complement A345170.
%Y A348609 A347438 counts factorizations with alternating product 1, additive A119620.
%Y A348609 A348379 counts factorizations w/ an alternating permutation, complement A348380.
%Y A348609 A348610 counts alternating ordered factorizations, complement A348613.
%Y A348609 Cf. A001248, A003242, A038548, A325534, A336103, A344614, A345162, A347050, A347437, A347706, A348377, A348381, A348382.
%K A348609 nonn
%O A348609 1,1
%A A348609 _Gus Wiseman_, Oct 30 2021