%I #15 Sep 06 2023 15:27:53
%S 0,0,0,1,1,2,4,5,7,11,16,20,28,37,50,65,84,106,140,175,222,277,350,
%T 432,539,663,819,999,1225,1489,1816,2192,2653,3191,3846,4603,5516,
%U 6578,7852,9327,11083,13120,15532,18328,21620,25430,29904,35071,41110,48080
%N Number of integer partitions of n of which every permutation has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.
%C Such a permutation is characterized by being neither a twin (x,x) nor wiggly (A025047, A345192). A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no wiggly permutations, even though it has the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).
%H Joseph Likar, <a href="/A344654/b344654.txt">Table of n, a(n) for n = 0..1000</a>
%e The a(3) = 1 through a(9) = 11 partitions:
%e (111) (1111) (2111) (222) (2221) (2222) (333)
%e (11111) (3111) (4111) (5111) (3222)
%e (21111) (31111) (41111) (6111)
%e (111111) (211111) (221111) (22221)
%e (1111111) (311111) (51111)
%e (2111111) (321111)
%e (11111111) (411111)
%e (2211111)
%e (3111111)
%e (21111111)
%e (111111111)
%t Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!MatchQ[#,{___,x_,y_,z_,___}/;x<=y<=z||x>=y>=z]&]=={}&]],{n,15}]
%Y The Heinz numbers of these partitions are A344653, complement A344742.
%Y The complement is counted by A344740.
%Y The normal case starts 0, 0, 0, then becomes A345162, complement A345163.
%Y Allowing twins (x,x) gives A345165, ranked by A345171.
%Y A001250 counts wiggly permutations.
%Y A003242 counts anti-run compositions.
%Y A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
%Y A325534 counts separable partitions, ranked by A335433.
%Y A325535 counts inseparable partitions, ranked by A335448.
%Y A344604 counts wiggly compositions with twins.
%Y A344605 counts wiggly patterns with twins.
%Y A344606 counts wiggly permutations of prime indices with twins.
%Y A344614 counts compositions with no consecutive strictly monotone triple.
%Y A345164 counts wiggly permutations of prime indices.
%Y A345170 counts partitions with a wiggly permutation, ranked by A345172.
%Y A345192 counts non-wiggly compositions.
%Y Cf. A000041, A000070, A102726, A103919, A333489, A335126, A344607, A344615, A345166, A345168, A345169.
%K nonn
%O 0,6
%A _Gus Wiseman_, Jun 12 2021
%E a(26)-a(32) from _Robert Price_, Jun 22 2021
%E a(33) onwards from _Joseph Likar_, Sep 06 2023