login
A350251
Number of non-alternating permutations of the multiset of prime factors of n.
8
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, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 2, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 8, 0, 0, 2, 1, 0, 2, 0, 2, 0, 2, 0, 9, 0, 0, 2, 2, 0, 2, 0, 5, 1, 0, 0, 8, 0, 0, 0
OFFSET
1,12
COMMENTS
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
FORMULA
a(n) = A008480(n) - A345164(n).
EXAMPLE
The a(n) permutations for selected n:
n = 4 12 24 48 60 72 90 96 120
----------------------------------------------------------------
22 223 2223 22223 2235 22233 2335 222223 22235
322 2232 22232 2253 22323 2353 222232 22253
2322 22322 2352 22332 2533 222322 22325
3222 23222 2532 23223 3235 223222 22352
32222 3225 23322 3325 232222 22523
3522 32223 3352 322222 22532
5223 32232 3532 23225
5322 32322 5233 23522
33222 5323 25223
5332 25322
32225
32252
32522
35222
52223
52232
52322
53222
MATHEMATICA
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]] ==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[n]]], !wigQ[#]&]], {n, 100}]
CROSSREFS
The non-anti-run case is A336107, complement A335452.
The complement is counted by A345164, with twins A344606.
Positions of nonzero terms are A345171, counted by A345165.
Positions of zeros are A345172, counted by A345170.
Compositions of this type are counted by A345192, ranked by A345168.
Ordered factorizations of this type counted by A348613, complement A348610.
Compositions weakly of this type are counted by A349053, ranked by A349057.
The weak version is A349797, complement A349056.
The case that is also weakly alternating is A349798, compositions A349800.
Patterns of this type are counted by A350252, complement A345194.
A001250 counts alternating permutations, complement A348615.
A003242 counts Carlitz (anti-run) compositions.
A008480 counts permutations of prime factors (ordered prime factorizations).
A025047/A025048/A025049 count alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798 (row lengths A001222).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344616 gives the alternating sum of prime indices, reverse A316524.
A349052/A129852/A129853 count weakly alternating compositions.
Sequence in context: A056674 A349798 A336107 * A367783 A363808 A227761
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 08 2022
STATUS
approved