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%I #12 Nov 05 2021 22:20:16
%S 270,378,594,702,918,1026,1242,1566,1620,1674,1750,1998,2214,2268,
%T 2322,2538,2625,2750,2862,3186,3250,3294,3564,3618,3834,3942,4050,
%U 4125,4212,4250,4266,4482,4750,4806,4875,5238,5454,5508,5562,5670,5750,5778,5886,6102
%N Numbers whose multiset of prime factors is separable but has no alternating permutation.
%C A multiset is separable if it has an anti-run permutation (no adjacent parts equal).
%C 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 has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%F Equals A345171 /\ A335433.
%e The terms together with their prime indices begin:
%e 270: {1,2,2,2,3}
%e 378: {1,2,2,2,4}
%e 594: {1,2,2,2,5}
%e 702: {1,2,2,2,6}
%e 918: {1,2,2,2,7}
%e 1026: {1,2,2,2,8}
%e 1242: {1,2,2,2,9}
%e 1566: {1,2,2,2,10}
%e 1620: {1,1,2,2,2,2,3}
%e 1674: {1,2,2,2,11}
%e 1750: {1,3,3,3,4}
%e 1998: {1,2,2,2,12}
%e 2214: {1,2,2,2,13}
%e 2268: {1,1,2,2,2,2,4}
%e 2322: {1,2,2,2,14}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t sepQ[y_]:=!MatchQ[y,{___,x_,x_,___}];
%t Select[Range[1000],Select[Permutations[primeMS[#]],wigQ]=={}&&!Select[Permutations[primeMS[#]],sepQ]=={}&]
%Y The partitions with these Heinz numbers are counted by A345166.
%Y Permutations of this type are ranked by A345169.
%Y Numbers with a factorization of this type are counted by A348609.
%Y A000041 counts integer partitions.
%Y A001250 counts alternating permutations, complement A348615.
%Y A003242 counts anti-run compositions.
%Y A025047 counts alternating compositions, ascend A025048, descend A025049.
%Y A325534 counts separable partitions, ranked by A335433.
%Y A325535 counts inseparable partitions, ranked by A335448.
%Y A344606 counts alternating permutations of prime indices with twins.
%Y A344740 counts twins and partitions with an alternating permutation.
%Y A345164 counts alternating permutations of prime factors.
%Y A345165 counts partitions without an alternating permutation.
%Y A345170 counts partitions with an alternating permutation.
%Y A345192 counts non-alternating compositions, without twins A348377.
%Y A348379 counts factorizations with an alternating permutation.
%Y Cf. A001222, A071321, A316524, A335126, A344614, A344616, A344652, A344653, A345163, A345168, A345193, A347706, A348380, A348613.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jun 13 2021
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