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Number of permutations of the prime indices of n with no adjacent triples (..., x, y, z, ...) such that x <= y <= z.
17

%I #7 Jun 18 2021 01:15:32

%S 1,1,1,1,1,2,1,0,1,2,1,2,1,2,2,0,1,2,1,2,2,2,1,1,1,2,0,2,1,5,1,0,2,2,

%T 2,3,1,2,2,1,1,5,1,2,2,2,1,0,1,2,2,2,1,1,2,1,2,2,1,7,1,2,2,0,2,5,1,2,

%U 2,5,1,2,1,2,2,2,2,5,1,0,0,2,1,7,2,2,2

%N Number of permutations of the prime indices of n with no adjacent triples (..., x, y, z, ...) such that x <= y <= z.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%e The permutations for n = 2, 6, 8, 30, 36, 60, 180, 210, 360:

%e (1) (12) (132) (1212) (1213) (12132) (1324) (121213)

%e (21) (213) (2121) (1312) (13212) (1423) (121312)

%e (231) (2211) (1321) (13221) (1432) (121321)

%e (312) (2131) (21213) (2143) (131212)

%e (321) (2311) (21312) (2314) (132121)

%e (3121) (21321) (2413) (132211)

%e (3211) (22131) (2431) (212131)

%e (23121) (3142) (213121)

%e (23211) (3214) (213211)

%e (31212) (3241) (221311)

%e (32121) (3412) (231211)

%e (32211) (3421) (312121)

%e (4132) (321211)

%e (4213)

%e (4231)

%e (4312)

%e (4321)

%t Table[Length[Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[n]]],!MatchQ[#,{___,x_,y_,z_,___}/;x<=y<=z]&]],{n,100}]

%Y All permutations of prime indices are counted by A008480.

%Y The case of permutations is A049774.

%Y Avoiding (3,2,1) also gives A344606.

%Y The wiggly case is A345164.

%Y A001250 counts wiggly permutations.

%Y A025047 counts wiggly compositions (ascend: A025048, descend: A025049).

%Y A056239 adds up prime indices, row sums of A112798.

%Y A325534 counts separable partitions, ranked by A335433.

%Y A325535 counts inseparable partitions, ranked by A335448.

%Y A335452 counts anti-run permutations of prime indices.

%Y A345170 counts partitions with a wiggly permutation, ranked by A345172.

%Y A345192 counts non-wiggly compositions, ranked by A345168.

%Y Counting compositions by patterns:

%Y - A102726 avoiding (1,2,3).

%Y - A128761 avoiding (1,2,3) adjacent.

%Y - A335514 matching (1,2,3).

%Y - A344614 avoiding (1,2,3) and (3,2,1) adjacent.

%Y - A344615 weakly avoiding (1,2,3) adjacent.

%Y Cf. A001222, A003242, A056986, A316524, A333213, A335511, A344604, A344653, A344654, A345167, A345173.

%K nonn

%O 1,6

%A _Gus Wiseman_, Jun 17 2021