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%I #22 Sep 06 2023 13:24:17
%S 0,0,1,1,2,2,5,5,8,11,17,20,29,37,51,65,85,106,141,175,223,277,351,
%T 432,540,663,820,999,1226,1489,1817,2192,2654,3191,3847,4603,5517,
%U 6578,7853,9327,11084,13120,15533,18328,21621,25430,29905,35071,41111,48080,56206,65554,76420,88918
%N Number of integer partitions of n without an alternating permutation.
%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).
%H Joseph Likar, <a href="/A345165/b345165.txt">Table of n, a(n) for n = 0..1000</a>
%H Joseph Likar, <a href="/A345165/a345165.java.txt">Java Implementation</a> using QBinomials
%e The a(2) = 1 through a(9) = 11 partitions:
%e (11) (111) (22) (2111) (33) (2221) (44) (333)
%e (1111) (11111) (222) (4111) (2222) (3222)
%e (3111) (31111) (5111) (6111)
%e (21111) (211111) (41111) (22221)
%e (111111) (1111111) (221111) (51111)
%e (311111) (321111)
%e (2111111) (411111)
%e (11111111) (2211111)
%e (3111111)
%e (21111111)
%e (111111111)
%t wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],wigQ]=={}&]],{n,0,15}]
%Y Excluding twins (x,x) gives A344654, complement A344740.
%Y The normal case is A345162, complement A345163.
%Y The complement is counted by A345170, ranked by A345172.
%Y The Heinz numbers of these partitions are A345171.
%Y The version for factorizations is A348380, complement A348379.
%Y A version for ordered factorizations is A348613, complement A348610.
%Y A000041 counts integer partitions.
%Y A001250 counts alternating permutations, complement A348615.
%Y A003242 counts anti-run compositions.
%Y A005649 counts anti-run patterns.
%Y A025047 counts alternating or wiggly compositions.
%Y A325534 counts separable partitions, ranked by A335433.
%Y A325535 counts inseparable partitions, ranked by A335448.
%Y A344604 counts alternating compositions with twins.
%Y A345164 counts alternating permutations of prime indices, w/ twins A344606.
%Y A345192 counts non-alternating compositions, without twins A348377.
%Y Cf. A000070, A025048, A025049, A103919, A335126, A344605, A344607, A344615, A344653, A345166, A345167, A345168, A345169, A347706, A348609.
%K nonn
%O 0,5
%A _Gus Wiseman_, Jun 12 2021
%E a(26) onwards by _Joseph Likar_, Aug 21 2023
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