login
Number of permutations of the prime indices of A000225(n) = 2^n - 1 with at least one non-singleton run.
1

%I #12 Sep 08 2020 02:19:52

%S 0,0,0,0,0,2,0,0,0,0,0,24,0,0,0,0,0,96,0,120,6,0,0,720,0,0,0,0,0,720,

%T 0,0,0,0,0,322560,0,0,0,5040,0,4320,0,0,0,0,0,362880,0,0

%N Number of permutations of the prime indices of A000225(n) = 2^n - 1 with at least one non-singleton run.

%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.

%F a(n) = A336107(2^n - 1).

%F a(n) = A336105(n) - A335432(n).

%e The a(21) = 6 permutations of {4, 4, 31, 68}:

%e (4,4,31,68)

%e (4,4,68,31)

%e (31,4,4,68)

%e (31,68,4,4)

%e (68,4,4,31)

%e (68,31,4,4)

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Length[Select[Permutations[primeMS[2^n-1]],MatchQ[#,{___,x_,x_,___}]&]],{n,30}]

%Y A335432 is the anti-run version.

%Y A335459 is the version for factorial numbers.

%Y A336105 counts all permutations of this multiset.

%Y A336107 is not restricted to predecessors of powers of 2.

%Y A003242 counts anti-run compositions.

%Y A005649 counts anti-run patterns.

%Y A008480 counts permutations of prime indices.

%Y A325534 counts separable partitions, ranked by A335433.

%Y A325535 counts inseparable partitions, ranked by A335448.

%Y A333489 ranks anti-run compositions.

%Y A335433 lists numbers whose prime indices have an anti-run permutation.

%Y A335448 lists numbers whose prime indices have no anti-run permutation.

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

%Y A335489 counts strict permutations of prime indices.

%Y Cf. A056239, A106351, A112798, A114938, A292884, A336102.

%Y The numbers 2^n - 1: A000225, A001265, A001348, A046051, A046800, A046801, A049093, A325610, A325611, A325612, A325625.

%K nonn,more

%O 1,6

%A _Gus Wiseman_, Sep 03 2020