A351202 Number of permutations of the multiset of prime factors of n (or ordered prime factorizations of n) with all distinct runs.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 2, 2, 1, 2, 2, 4, 1, 6, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 2, 1, 2, 6, 1, 2, 2, 6, 1, 4, 1, 2, 2, 2, 2, 6, 1, 4, 1, 2, 1, 6, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Feb 13 2022

Keywords

• nonn

Examples

			The a(36) = 2 permutations are (1,1,2,2), (2,2,1,1). Missing are: (1,2,1,2), (1,2,2,1), (2,1,1,2), (2,1,2,1). Here we use prime indices instead of factors.
		

Links

• Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Crossrefs

The maximum number of possible permutations is A008480.

Positions less than A008480 are A351201.

A005811 counts runs in binary expansion.

A044813 lists numbers whose binary expansion has distinct run-lengths.

A056239 adds up prime indices, row sums of A112798.

A098859 counts partitions with distinct multiplicities, ordered A242882.

A283353 counts normal multisets with a permutation without distinct runs.

A297770 counts distinct runs in binary expansion.

A351014 counts distinct runs in standard compositions, firsts A351015.

A351204 = partitions whose perms. have distinct runs, complement A351203.

Counting words with all distinct runs:

- A351013 = compositions, for run-lengths A329739, ranked by A351290.

- A351016 = binary words, for run-lengths A351017.

- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.

- A351200 = patterns, for run-lengths A351292.

Cf. A001055, A001221, A001222, A003242, A061395, A116608, A238130 or A238279, A328592, A351291.

Programs

Mathematica

Table[Length[Select[Permutations[Join@@ ConstantArray@@@FactorInteger[n]],UnsameQ@@Split[#]&]],{n,100}]