cp's OEIS Frontend

A332742 Number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.

%I A332742 #6 Feb 16 2025 08:33:59
%S A332742 0,0,0,0,0,1,0,2,3,2,0,8,0,3,7,16,0,24,0,16,12,4,0,52,16,5,81,26,0,54,
%T A332742 0,104,18,6,31,168,0,7,25,112,0,99,0,38,201,8,0,344,65,132,33,52,0,
%U A332742 612,52,202,42,9,0,408,0,10,411,688,80,162,0,68,52,272
%N A332742 Number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.
%C A332742 This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%C A332742 A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332742 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F A332742 a(n) + A332741(n) = A318762(n).
%e A332742 The a(n) permutations for n = 6, 8, 9, 10, 12, 14, 15, 16:
%e A332742   121  132  1212  1121  1132  11121  11212  1243
%e A332742        231  1221  1211  1213  11211  11221  1324
%e A332742             2121        1231  12111  12112  1342
%e A332742                         1312         12121  1423
%e A332742                         1321         12211  1432
%e A332742                         2131         21121  2143
%e A332742                         2311         21211  2314
%e A332742                         3121                2341
%e A332742                                             2413
%e A332742                                             2431
%e A332742                                             3142
%e A332742                                             3241
%e A332742                                             3412
%e A332742                                             3421
%e A332742                                             4132
%e A332742                                             4231
%t A332742 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A332742 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
%t A332742 Table[Length[Select[Permutations[nrmptn[n]],!unimodQ[#]&]],{n,30}]
%Y A332742 Dominated by A318762.
%Y A332742 The complement of the non-negated version is counted by A332294.
%Y A332742 The non-negated version is A332672.
%Y A332742 The complement is counted by A332741.
%Y A332742 A less interesting version is A333146.
%Y A332742 Unimodal compositions are A001523.
%Y A332742 Unimodal normal sequences are A007052.
%Y A332742 Non-unimodal normal sequences are A328509.
%Y A332742 Partitions with non-unimodal 0-appended first differences are A332284.
%Y A332742 Compositions whose negation is unimodal are A332578.
%Y A332742 Partitions with non-unimodal negated run-lengths are A332639.
%Y A332742 Numbers whose negated prime signature is not unimodal are A332642.
%Y A332742 Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332283, A332288, A332638, A332669, A333145.
%K A332742 nonn
%O A332742 1,8
%A A332742 _Gus Wiseman_, Mar 09 2020