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A332288 Number of unimodal permutations of the multiset of prime indices of n.

%I A332288 #6 Feb 16 2025 08:33:59
%S A332288 1,1,1,1,1,2,1,1,1,2,1,3,1,2,2,1,1,2,1,3,2,2,1,4,1,2,1,3,1,4,1,1,2,2,
%T A332288 2,3,1,2,2,4,1,4,1,3,3,2,1,5,1,2,2,3,1,2,2,4,2,2,1,6,1,2,3,1,2,4,1,3,
%U A332288 2,4,1,4,1,2,2,3,2,4,1,5,1,2,1,6,2,2,2
%N A332288 Number of unimodal permutations of the multiset of prime indices of n.
%C A332288 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.
%C A332288 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%C A332288 Also permutations of the multiset of prime indices of n avoiding the patterns (2,1,2), (2,1,3), and (3,1,2).
%H A332288 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A332288 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332288 The a(n) permutations for n = 2, 6, 12, 24, 48, 60, 120, 180:
%e A332288   (1)  (12)  (112)  (1112)  (11112)  (1123)  (11123)  (11223)
%e A332288        (21)  (121)  (1121)  (11121)  (1132)  (11132)  (11232)
%e A332288              (211)  (1211)  (11211)  (1231)  (11231)  (11322)
%e A332288                     (2111)  (12111)  (1321)  (11321)  (12231)
%e A332288                             (21111)  (2311)  (12311)  (12321)
%e A332288                                      (3211)  (13211)  (13221)
%e A332288                                              (23111)  (22311)
%e A332288                                              (32111)  (23211)
%e A332288                                                       (32211)
%t A332288 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A332288 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332288 Table[Length[Select[Permutations[primeMS[n]],unimodQ]],{n,30}]
%Y A332288 Dominated by A008480.
%Y A332288 A more interesting version is A332294.
%Y A332288 The complement is counted by A332671.
%Y A332288 Unimodal compositions are A001523.
%Y A332288 Unimodal normal sequences appear to be A007052.
%Y A332288 Unimodal permutations are A011782.
%Y A332288 Non-unimodal permutations are A059204.
%Y A332288 Numbers with non-unimodal unsorted prime signature are A332282.
%Y A332288 Partitions with unimodal 0-appended first differences are A332283.
%Y A332288 Cf. A056239, A112798, A115981, A124010, A227038, A304660, A328509, A332280, A332284, A332294, A332578, A332672.
%K A332288 nonn
%O A332288 1,6
%A A332288 _Gus Wiseman_, Feb 22 2020