A344652 Number of permutations of the prime indices of n with no adjacent triples (..., x, y, z, ...) such that x <= y <= z.

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, 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, 2, 5, 1, 2, 1, 2, 2, 2, 2, 5, 1, 0, 0, 2, 1, 7, 2, 2, 2

Offset

1,6

Comments

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.

Links

Table of n, a(n) for n=1..87.

Example

The permutations for n = 2, 6, 8, 30, 36, 60, 180, 210, 360:
  (1)  (12)  (132)  (1212)  (1213)  (12132)  (1324)  (121213)
       (21)  (213)  (2121)  (1312)  (13212)  (1423)  (121312)
             (231)  (2211)  (1321)  (13221)  (1432)  (121321)
             (312)          (2131)  (21213)  (2143)  (131212)
             (321)          (2311)  (21312)  (2314)  (132121)
                            (3121)  (21321)  (2413)  (132211)
                            (3211)  (22131)  (2431)  (212131)
                                    (23121)  (3142)  (213121)
                                    (23211)  (3214)  (213211)
                                    (31212)  (3241)  (221311)
                                    (32121)  (3412)  (231211)
                                    (32211)  (3421)  (312121)
                                             (4132)  (321211)
                                             (4213)
                                             (4231)
                                             (4312)
                                             (4321)

Mathematica

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

Crossrefs

All permutations of prime indices are counted by A008480.

The case of permutations is A049774.

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

The wiggly case is A345164.

A001250 counts wiggly permutations.

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

A056239 adds up prime indices, row sums of A112798.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A335452 counts anti-run permutations of prime indices.

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

A345192 counts non-wiggly compositions, ranked by A345168.

Counting compositions by patterns:

- A102726 avoiding (1,2,3).

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

- A335514 matching (1,2,3).

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

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

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

Sequence in context: A087991 A335451 A366078 * A366074 A293439 A144095

Adjacent sequences: A344649 A344650 A344651 * A344653 A344654 A344655

Keyword

nonn

Author

Gus Wiseman, Jun 17 2021

Status

approved