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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.
37

%I #18 Sep 06 2023 13:25:03

%S 1,1,2,2,4,5,7,10,15,19,26,36,49,64,85,111,147,191,245,315,405,515,

%T 652,823,1036,1295,1617,2011,2493,3076,3788,4650,5696,6952,8464,10280,

%U 12461,15059,18163,21858,26255,31463,37642,44933,53555,63704,75654,89683,106163,125445,148021

%N 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 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 Joseph Likar, <a href="/A344740/b344740.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A345170(n) for n odd; a(n) = A345170(n) + 1 for n even.

%e The a(1) = 1 through a(8) = 15 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (1,1) (2,1) (2,2) (3,2) (3,3) (4,3) (4,4)

%e (3,1) (4,1) (4,2) (5,2) (5,3)

%e (2,1,1) (2,2,1) (5,1) (6,1) (6,2)

%e (3,1,1) (3,2,1) (3,2,2) (7,1)

%e (4,1,1) (3,3,1) (3,3,2)

%e (2,2,1,1) (4,2,1) (4,2,2)

%e (5,1,1) (4,3,1)

%e (3,2,1,1) (5,2,1)

%e (2,2,1,1,1) (6,1,1)

%e (3,2,2,1)

%e (3,3,1,1)

%e (4,2,1,1)

%e (2,2,2,1,1)

%e (3,2,1,1,1)

%e 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 Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!MatchQ[#,{___,x_,y_,z_,___}/;x<=y<=z||x>=y>=z]&]!={}&]],{n,0,15}]

%Y The complement is counted by A344654.

%Y The Heinz numbers of these partitions are A344742, complement A344653.

%Y The normal case starts 1, 1, 1, then becomes A345163, complement A345162.

%Y Not counting twins (x,x) gives A345170, ranked by A345172.

%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 A345165 counts partitions without a wiggly permutation, ranked by A345171.

%Y A345192 counts non-wiggly compositions.

%Y Cf. A000041, A000070, A102726, A103919, A333489, A344607, A344612, A344615, A345166, A345168, A345169.

%K nonn

%O 0,3

%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 05 2023