This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350251 #8 Jan 18 2022 05:56:13
%S A350251 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,
%T A350251 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,
%U A350251 0,2,0,9,0,0,2,2,0,2,0,5,1,0,0,8,0,0,0
%N A350251 Number of non-alternating permutations of the multiset of prime factors of n.
%C A350251 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).
%F A350251 a(n) = A008480(n) - A345164(n).
%e A350251 The a(n) permutations for selected n:
%e A350251 n = 4 12 24 48 60 72 90 96 120
%e A350251 ----------------------------------------------------------------
%e A350251 22 223 2223 22223 2235 22233 2335 222223 22235
%e A350251 322 2232 22232 2253 22323 2353 222232 22253
%e A350251 2322 22322 2352 22332 2533 222322 22325
%e A350251 3222 23222 2532 23223 3235 223222 22352
%e A350251 32222 3225 23322 3325 232222 22523
%e A350251 3522 32223 3352 322222 22532
%e A350251 5223 32232 3532 23225
%e A350251 5322 32322 5233 23522
%e A350251 33222 5323 25223
%e A350251 5332 25322
%e A350251 32225
%e A350251 32252
%e A350251 32522
%e A350251 35222
%e A350251 52223
%e A350251 52232
%e A350251 52322
%e A350251 53222
%t A350251 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]] ==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A350251 Table[Length[Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[n]]],!wigQ[#]&]],{n,100}]
%Y A350251 The non-anti-run case is A336107, complement A335452.
%Y A350251 The complement is counted by A345164, with twins A344606.
%Y A350251 Positions of nonzero terms are A345171, counted by A345165.
%Y A350251 Positions of zeros are A345172, counted by A345170.
%Y A350251 Compositions of this type are counted by A345192, ranked by A345168.
%Y A350251 Ordered factorizations of this type counted by A348613, complement A348610.
%Y A350251 Compositions weakly of this type are counted by A349053, ranked by A349057.
%Y A350251 The weak version is A349797, complement A349056.
%Y A350251 The case that is also weakly alternating is A349798, compositions A349800.
%Y A350251 Patterns of this type are counted by A350252, complement A345194.
%Y A350251 A001250 counts alternating permutations, complement A348615.
%Y A350251 A003242 counts Carlitz (anti-run) compositions.
%Y A350251 A008480 counts permutations of prime factors (ordered prime factorizations).
%Y A350251 A025047/A025048/A025049 count alternating compositions, ranked by A345167.
%Y A350251 A056239 adds up prime indices, row sums of A112798 (row lengths A001222).
%Y A350251 A325534 counts separable partitions, ranked by A335433.
%Y A350251 A325535 counts inseparable partitions, ranked by A335448.
%Y A350251 A344616 gives the alternating sum of prime indices, reverse A316524.
%Y A350251 A349052/A129852/A129853 count weakly alternating compositions.
%Y A350251 Cf. A071321, A289553, A316523, A344652, A344653, A344742, A345173, A348379, A349061, A349801, A350138, A350139.
%K A350251 nonn
%O A350251 1,12
%A A350251 _Gus Wiseman_, Jan 08 2022