This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A344740 #18 Sep 06 2023 13:25:03
%S A344740 1,1,2,2,4,5,7,10,15,19,26,36,49,64,85,111,147,191,245,315,405,515,
%T A344740 652,823,1036,1295,1617,2011,2493,3076,3788,4650,5696,6952,8464,10280,
%U A344740 12461,15059,18163,21858,26255,31463,37642,44933,53555,63704,75654,89683,106163,125445,148021
%N A344740 Number of integer partitions of n with a permutation that has no consecutive monotone triple, i.e., no triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.
%C A344740 These partitions are characterized by either being a twin (x,x) or having a wiggly permutation. A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no wiggly permutations, even though it has anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
%H A344740 Joseph Likar, <a href="/A344740/b344740.txt">Table of n, a(n) for n = 0..1000</a>
%F A344740 a(n) = A345170(n) for n odd; a(n) = A345170(n) + 1 for n even.
%e A344740 The a(1) = 1 through a(8) = 15 partitions:
%e A344740 (1) (2) (3) (4) (5) (6) (7) (8)
%e A344740 (1,1) (2,1) (2,2) (3,2) (3,3) (4,3) (4,4)
%e A344740 (3,1) (4,1) (4,2) (5,2) (5,3)
%e A344740 (2,1,1) (2,2,1) (5,1) (6,1) (6,2)
%e A344740 (3,1,1) (3,2,1) (3,2,2) (7,1)
%e A344740 (4,1,1) (3,3,1) (3,3,2)
%e A344740 (2,2,1,1) (4,2,1) (4,2,2)
%e A344740 (5,1,1) (4,3,1)
%e A344740 (3,2,1,1) (5,2,1)
%e A344740 (2,2,1,1,1) (6,1,1)
%e A344740 (3,2,2,1)
%e A344740 (3,3,1,1)
%e A344740 (4,2,1,1)
%e A344740 (2,2,2,1,1)
%e A344740 (3,2,1,1,1)
%e A344740 For example, the partition (3,2,2,1) has the two wiggly permutations (2,3,1,2) and (2,1,3,2), so is counted under a(8).
%t A344740 Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!MatchQ[#,{___,x_,y_,z_,___}/;x<=y<=z||x>=y>=z]&]!={}&]],{n,0,15}]
%Y A344740 The complement is counted by A344654.
%Y A344740 The Heinz numbers of these partitions are A344742, complement A344653.
%Y A344740 The normal case starts 1, 1, 1, then becomes A345163, complement A345162.
%Y A344740 Not counting twins (x,x) gives A345170, ranked by A345172.
%Y A344740 A001250 counts wiggly permutations.
%Y A344740 A003242 counts anti-run compositions.
%Y A344740 A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
%Y A344740 A325534 counts separable partitions, ranked by A335433.
%Y A344740 A325535 counts inseparable partitions, ranked by A335448.
%Y A344740 A344604 counts wiggly compositions with twins.
%Y A344740 A344605 counts wiggly patterns with twins.
%Y A344740 A344606 counts wiggly permutations of prime indices with twins.
%Y A344740 A344614 counts compositions with no consecutive strictly monotone triple.
%Y A344740 A345164 counts wiggly permutations of prime indices.
%Y A344740 A345165 counts partitions without a wiggly permutation, ranked by A345171.
%Y A344740 A345192 counts non-wiggly compositions.
%Y A344740 Cf. A000041, A000070, A102726, A103919, A333489, A344607, A344612, A344615, A345166, A345168, A345169.
%K A344740 nonn
%O A344740 0,3
%A A344740 _Gus Wiseman_, Jun 12 2021
%E A344740 a(26)-a(32) from _Robert Price_, Jun 22 2021
%E A344740 a(33) onwards from _Joseph Likar_, Sep 05 2023