A335489 -id:A335489 - cp's OEIS frontend

A335407 Number of anti-run permutations of the prime indices of n!.

1, 1, 1, 2, 0, 2, 3, 54, 0, 30, 105, 6090, 1512, 133056, 816480, 127209600, 0, 10090080, 562161600, 69864795000, 49989139200, 29593652088000, 382147120555200, 41810689605484800, 4359985823793600, 3025062801079038720, 49052072750637116160, 25835971971637227375360

Offset: 0

Gus Wiseman, Jul 01 2020

nonn

An anti-run is a sequence with no adjacent equal parts.
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.
Conjecture: Only vanishes at n = 4 and n = 8.
a(16) = 0. Proof: 16! = 2^15 * m where bigomega(m) = A001222(m) = 13. We can't separate 15 1's with 13 other numbers. - David A. Corneth, Jul 04 2020

			The a(0) = 1 through a(6) = 3 anti-run permutations:
  ()  ()  (1)  (1,2)  .  (1,2,1,3,1)  (1,2,1,2,1,3,1)
               (2,1)     (1,3,1,2,1)  (1,2,1,3,1,2,1)
                                      (1,3,1,2,1,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version for Mersenne numbers is A335432.
Anti-run compositions are A003242.
Anti-run patterns are counted by A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Factorial numbers: A000142, A001222, A002982, A007489, A022559, A027423, A054991, A108731, A181819, A181821, A325272, A325273, A325617.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n!]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,10}]

\\ See A335452 for count.
a(n)={count(factor(n!)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000142(n)). - Andrew Howroyd, Feb 03 2021

Terms a(14) and beyond from Andrew Howroyd, Feb 03 2021

A336107 Number of permutations of the prime indices of n with at least one non-singleton run, or non-separations.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 6, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 9, 0, 0, 2, 2, 0, 0, 0, 5, 1, 0, 0, 6, 0, 0, 0

Offset: 1

Gus Wiseman, Sep 03 2020

nonn

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.

A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.

			The a(n) non-separations for n = 12, 36, 60, 72, 180, 420:
  (11)  (112)  (1122)  (1123)  (11122)  (11223)  (11234)
        (211)  (1221)  (1132)  (11212)  (11232)  (11243)
               (2112)  (2113)  (11221)  (11322)  (11324)
               (2211)  (2311)  (12112)  (12213)  (11342)
                       (3112)  (12211)  (12231)  (11423)
                       (3211)  (21112)  (13122)  (11432)
                               (21121)  (13221)  (21134)
                               (21211)  (21123)  (21143)
                               (22111)  (21132)  (23114)
                                        (22113)  (23411)
                                        (22131)  (24113)
                                        (22311)  (24311)
                                        (23112)  (31124)
                                        (23211)  (31142)
                                        (31122)  (32114)
                                        (31221)  (32411)
                                        (32112)  (34112)
                                        (32211)  (34211)
                                                 (41123)
                                                 (41132)
                                                 (42113)
                                                 (42311)
                                                 (43112)
                                                 (43211)

A005117 lists positions of zeros, with complement A013929.
A008480 counts permutations of prime indices, ranked by A333221.
A003242 and A335452 count separations, ranked by A333489.
A325535 counts inseparable partitions, ranked by A335448.
A325534 counts separable partitions, ranked by A335433.
Cf. A056239, A112798, A261962, A335451, A335452, A335460, A335489.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,x_,_}]&]],{n,100}]

a(n) = A008480(n) - A335452(n).
a(A000961(n)) = 0 if n is in A027883, otherwise 1.
a(A005117(n)) = 0.
a(n!) = A335459(n).
a(A006939(n)) = A022915(n).

A348381 Number of inseparable factorizations of n that are not a twin (x*x).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 30 2021

nonn

First differs from A347706 at a(216) = 3, A347706(216) = 4.
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
A multiset is inseparable if it has no permutation that is an anti-run, meaning there are always adjacent equal parts. Alternatively, a multiset is inseparable if its maximal multiplicity is at most one plus the sum of its remaining multiplicities.

			The a(n) factorizations for n = 96, 192, 384, 576:
  2*2*2*12      3*4*4*4         4*4*4*6           4*4*4*9
  2*2*2*2*6     2*2*2*24        2*2*2*48          2*2*2*72
  2*2*2*2*2*3   2*2*2*2*12      2*2*2*2*24        2*2*2*2*36
                2*2*2*2*2*6     2*2*2*2*3*8       2*2*2*2*4*9
                2*2*2*2*3*4     2*2*2*2*4*6       2*2*2*2*6*6
                2*2*2*2*2*2*3   2*2*2*2*2*12      2*2*2*2*2*18
                                2*2*2*2*2*2*6     2*2*2*2*3*12
                                2*2*2*2*2*3*4     2*2*2*2*2*2*9
                                2*2*2*2*2*2*2*3   2*2*2*2*2*3*6
                                                  2*2*2*2*2*2*3*3

Positions of nonzero terms are A046099.
Partitions not of this type are counted by A325534 - A000035.
Partitions of this type are counted by A325535 - A000035.
Allowing twins gives A333487.
The case without an alternating permutation is A347706, with twins A348380.
The complement is counted by A348383, without twins A335434.
A001055 counts factorizations, strict A045778, ordered A074206.
A001250 counts alternating permutations of sets.
A008480 counts permutations of prime indices, strict A335489.
A025047 counts alternating or wiggly compositions.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A344654 counts non-twin partitions without an alternating permutation.
A348382 counts non-anti-run compositions that are not a twin.
A348611 counts anti-run ordered factorizations.
Cf. A038548, A336102, A344614, A344653, A344740, A347050, A347437, A347438, A347456, A348379, A348609.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],!MatchQ[#,{x_,x_}]&&Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]=={}&]],{n,100}]

a(n > 1) = A333487(n) - A010052(n).

a(2^n) = A325535(n) - 1 for odd n, otherwise A325535(n).

A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 6, 2, 36, 1, 6, 6, 24, 1, 24, 1, 240, 6, 24, 2, 1800, 6, 6, 6, 720, 6, 1800, 1, 120, 24, 6, 24, 282240, 2, 6, 24, 15120, 2, 5760, 6, 5040, 720, 24, 6, 1451520, 2, 5040, 120, 5040, 6, 1800, 720, 40320, 24, 720, 2, 1117670400, 1, 6, 1800, 5040, 6

Offset: 1

Gus Wiseman, Jul 02 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

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.

			The a(1) = 1 through a(10) = 6 permutations:
  ()  (2)  (4)  (2,3)  (11)  (2,4,2)  (31)  (2,3,7)  (21,4)  (11,2,5)
                (3,2)                       (2,7,3)  (4,21)  (11,5,2)
                                            (3,2,7)          (2,11,5)
                                            (3,7,2)          (2,5,11)
                                            (7,2,3)          (5,11,2)
                                            (7,3,2)          (5,2,11)

Andrew Howroyd, Table of n, a(n) for n = 1..200

The version for factorial numbers is A335407.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Permutations of prime indices are A008480.
Anti-runs are ranked by A333489.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Strict permutations of prime indices are A335489.
Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.
Mersenne numbers: A000225, A046051, A046800, A046801, A049093, A059305, A159611, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[2^n-1]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,30}]

\\ See A335452 for count.
a(n) = {count(factor(2^n-1)[,2])} \\ Andrew Howroyd, Feb 03 2021

a(n) = A335452(A000225(n)).

Terms a(51) and beyond from Andrew Howroyd, Feb 03 2021

A335487 Number of (1,1)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 3, 0, 0, 0, 4, 1, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 3, 3, 0, 0, 5, 1, 3, 0, 3, 0, 4, 0, 4, 0, 0, 0, 12, 0, 0, 3, 1, 0, 0, 0, 3, 0, 0, 0, 10, 0, 0, 3, 3, 0, 0, 0, 5, 1, 0, 0, 12, 0, 0

Offset: 1

Gus Wiseman, Jun 14 2020

nonn

Depends only on sorted prime signature (A118914).
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.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(n) permutations for n = 4, 12, 24, 48, 36, 72, 60:
  (11)  (112)  (1112)  (11112)  (1122)  (11122)  (1123)
        (121)  (1121)  (11121)  (1212)  (11212)  (1132)
        (211)  (1211)  (11211)  (1221)  (11221)  (1213)
               (2111)  (12111)  (2112)  (12112)  (1231)
                       (21111)  (2121)  (12121)  (1312)
                                (2211)  (12211)  (1321)
                                        (21112)  (2113)
                                        (21121)  (2131)
                                        (21211)  (2311)
                                        (22111)  (3112)
                                                 (3121)
                                                 (3211)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Positions of zeros are A005117 (squarefree numbers).
The case where the match must be contiguous is A333175.
The avoiding version is A335489.
The (1,1,1)-matching case is A335510.
Patterns are counted by A000670.
Permutations of prime indices are counted by A008480.
(1,1)-matching patterns are counted by A019472.
(1,1)-matching compositions are counted by A261982.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
Dimensions of downsets of standard compositions are A335465.
(1,1)-matching compositions are ranked by A335488.
Cf. A000961, A056239, A056986, A112798, A181796, A335451, A335452, A335460, A335462.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!UnsameQ@@#&]],{n,100}]

a(n) = 0 if n is squarefree, otherwise a(n) = A008480(n).

a(n) = A008480(n) - A281188(n) for n != 4.

A336104 Number of permutations of the prime indices of A000225(n) = 2^n - 1 with at least one non-singleton run.

Original entry on oeis.org

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, 0, 0, 0, 0, 0, 322560, 0, 0, 0, 5040, 0, 4320, 0, 0, 0, 0, 0, 362880, 0, 0

Offset: 1

Gus Wiseman, Sep 03 2020

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.

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

A335432 is the anti-run version.
A335459 is the version for factorial numbers.
A336105 counts all permutations of this multiset.
A336107 is not restricted to predecessors of powers of 2.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A008480 counts permutations of prime indices.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A333489 ranks anti-run compositions.
A335433 lists numbers whose prime indices have an anti-run permutation.
A335448 lists numbers whose prime indices have no anti-run permutation.
A335452 counts anti-run permutations of prime indices.
A335489 counts strict permutations of prime indices.
Cf. A056239, A106351, A112798, A114938, A292884, A336102.
The numbers 2^n - 1: A000225, A001265, A001348, A046051, A046800, A046801, A049093, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[2^n-1]],MatchQ[#,{_,x_,x_,_}]&]],{n,30}]

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

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

A336105 Number of permutations of the prime indices of 2^n - 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 6, 2, 6, 2, 60, 1, 6, 6, 24, 1, 120, 1, 360, 12, 24, 2, 2520, 6, 6, 6, 720, 6, 2520, 1, 120, 24, 6, 24, 604800, 2, 6, 24, 20160, 2, 10080, 6, 5040, 720, 24, 6, 1814400, 2, 5040, 120, 5040, 6, 15120, 720, 40320, 24, 720, 2

Offset: 1

Gus Wiseman, Sep 03 2020

nonn

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.

			The a(n) permutations for n = 2, 4, 6, 8, 21:
  (2)  (2,3)  (2,2,4)  (2,3,7)  (31,4,4,68)
       (3,2)  (2,4,2)  (2,7,3)  (31,4,68,4)
              (4,2,2)  (3,2,7)  (31,68,4,4)
                       (3,7,2)  (4,31,4,68)
                       (7,2,3)  (4,31,68,4)
                       (7,3,2)  (4,4,31,68)
                                (4,4,68,31)
                                (4,68,31,4)
                                (4,68,4,31)
                                (68,31,4,4)
                                (68,4,31,4)
                                (68,4,4,31)

A008480 is not restricted to predecessors of powers of 2.
A325617 is the version for factorial numbers.
A335489 counts strict permutations of prime indices.
Cf. A056239, A112798, A335432, A336104.
The numbers 2^n - 1: A000225, A046051, A046800, A046801, A049093, A325610, A325611, A325612, A325625.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Permutations[primeMS[2^n-1]]],{n,30}]

a(n) = A008480(2^n - 1).

a(n) = A336104(n) + A335432(n).

cp's OEIS Frontend

A335407 Number of anti-run permutations of the prime indices of n!.

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A336107 Number of permutations of the prime indices of n with at least one non-singleton run, or non-separations.

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A348381 Number of inseparable factorizations of n that are not a twin (x*x).

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A335432 Number of anti-run permutations of the prime indices of Mersenne numbers A000225(n) = 2^n - 1.

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A335487 Number of (1,1)-matching permutations of the prime indices of n.

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A336104 Number of permutations of the prime indices of A000225(n) = 2^n - 1 with at least one non-singleton run.

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A336105 Number of permutations of the prime indices of 2^n - 1.

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