cp's OEIS Frontend

A345164 Number of alternating permutations of the multiset of prime factors of n.

%I A345164 #16 Nov 15 2021 01:17:53
%S A345164 1,1,1,0,1,2,1,0,0,2,1,1,1,2,2,0,1,1,1,1,2,2,1,0,0,2,0,1,1,4,1,0,2,2,
%T A345164 2,2,1,2,2,0,1,4,1,1,1,2,1,0,0,1,2,1,1,0,2,0,2,2,1,4,1,2,1,0,2,4,1,1,
%U A345164 2,4,1,1,1,2,1,1,2,4,1,0,0,2,1,4,2,2,2
%N A345164 Number of alternating permutations of the multiset of prime factors of n.
%C A345164 First differs from A335452 at a(30) = 4, A335452(30) = 6. The anti-runs (2,3,5) and (5,3,2) are not alternating.
%C A345164 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 permutation, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
%e A345164 The a(n) alternating permutations of prime indices for n = 180, 210, 300, 420, 900:
%e A345164   (12132)  (1324)  (13132)  (12143)  (121323)
%e A345164   (21213)  (1423)  (13231)  (13142)  (132312)
%e A345164   (21312)  (2143)  (21313)  (13241)  (213132)
%e A345164   (23121)  (2314)  (23131)  (14132)  (213231)
%e A345164   (31212)  (2413)  (31213)  (14231)  (231213)
%e A345164            (3142)  (31312)  (21314)  (231312)
%e A345164            (3241)           (21413)  (312132)
%e A345164            (3412)           (23141)  (323121)
%e A345164            (4132)           (24131)
%e A345164            (4231)           (31214)
%e A345164                             (31412)
%e A345164                             (34121)
%e A345164                             (41213)
%e A345164                             (41312)
%t A345164 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A345164 Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],wigQ]],{n,30}]
%Y A345164 Counting all permutations gives A008480.
%Y A345164 Dominated by A335452 (number of separations of prime factors).
%Y A345164 Including twins (x,x) gives A344606.
%Y A345164 Positions of zeros are A345171, counted by A345165.
%Y A345164 Positions of nonzero terms are A345172.
%Y A345164 A000041 counts integer partitions.
%Y A345164 A001250 counts alternating permutations.
%Y A345164 A003242 counts anti-run compositions.
%Y A345164 A025047 counts alternating or wiggly compositions, also A025048, A025049.
%Y A345164 A325534 counts separable partitions, ranked by A335433.
%Y A345164 A325535 counts inseparable partitions, ranked by A335448.
%Y A345164 A344604 counts alternating compositions with twins.
%Y A345164 A344654 counts non-twin partitions w/o alternating permutation, rank: A344653.
%Y A345164 A344740 counts twins and partitions w/ alternating permutation, rank: A344742.
%Y A345164 A345166 counts separable partitions w/o alternating permutation, rank: A345173.
%Y A345164 A345170 counts partitions with a alternating permutation.
%Y A345164 Cf. A001222, A071321, A071322, A316523, A316524, A333489, A335126, A344605, A344614, A344616, A344652, A345163, A345168.
%K A345164 nonn
%O A345164 1,6
%A A345164 _Gus Wiseman_, Jun 13 2021