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A344606 Number of alternating permutations of the prime factors of n, counting multiplicity, including twins (x,x).

%I A344606 #15 Dec 21 2022 20:11:22
%S A344606 1,1,1,1,1,2,1,0,1,2,1,1,1,2,2,0,1,1,1,1,2,2,1,0,1,2,0,1,1,4,1,0,2,2,
%T A344606 2,2,1,2,2,0,1,4,1,1,1,2,1,0,1,1,2,1,1,0,2,0,2,2,1,4,1,2,1,0,2,4,1,1,
%U A344606 2,4,1,1,1,2,1,1,2,4,1,0,0,2,1,4,2,2,2
%N A344606 Number of alternating permutations of the prime factors of n, counting multiplicity, including twins (x,x).
%C A344606 Differs from A335448 in having a(x^2) = 0 and a(270) = 0.
%C A344606 These are permutations of the prime factors of n, counting multiplicity, with no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z.
%C A344606 The version without twins (x,x) is A345164, which is identical to this sequence except when n is the square of a prime.
%e A344606 The permutations for n = 2, 6, 30, 180, 210, 300, 420, 720, 840:
%e A344606   2   23   253   23253   2537   25253   23275   2323252   232527
%e A344606       32   325   32325   2735   25352   25273   2325232   232725
%e A344606            352   32523   3275   32525   25372   2523232   252327
%e A344606            523   35232   3527   35252   27253             252723
%e A344606                  52323   3725   52325   27352             272325
%e A344606                          5273   52523   32527             272523
%e A344606                          5372           32725             325272
%e A344606                          5723           35272             327252
%e A344606                          7253           37252             523272
%e A344606                          7352           52327             527232
%e A344606                                         52723             723252
%e A344606                                         57232             725232
%e A344606                                         72325
%e A344606                                         72523
%e A344606 For example, there are no alternating permutations of the prime factors of 270 because the only anti-runs are {3,2,3,5,3} and {3,5,3,2,3}, neither of which is alternating, so a(270) = 0.
%t A344606 Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],!MatchQ[#,{___,x_,y_,z_,___}/;x<=y<=z||x>=y>=z]&]],{n,100}]
%Y A344606 The version for permutations is A001250.
%Y A344606 The extension to anti-run permutations is A335452.
%Y A344606 The version for compositions is A344604.
%Y A344606 The version for patterns is A344605.
%Y A344606 Positions of zeros are A344653 (counted by A344654).
%Y A344606 Not including twins (x,x) gives A345164.
%Y A344606 A008480 counts permutations of prime indices (strict: A335489, rank: A333221).
%Y A344606 A056239 adds up prime indices,  row sums of A112798.
%Y A344606 A071321 and A071322 are signed sums of prime factors.
%Y A344606 A316523 is a signed sum of prime multiplicities.
%Y A344606 A316524 and A344616 are signed sums of prime indices.
%Y A344606 A325534 counts separable partitions (ranked by A335433).
%Y A344606 A325535 counts inseparable partitions (ranked by A335448).
%Y A344606 A344740 counts partitions with an alternating permutation or twin (x,x).
%Y A344606 Cf. A000041, A000607, A000961, A001222, A003242, A026424, A028260, A049774, A103919, A181796, A343938, A344652.
%K A344606 nonn
%O A344606 1,6
%A A344606 _Gus Wiseman_, May 28 2021