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.

1, 1, 2, 2, 4, 5, 7, 10, 15, 19, 26, 36, 49, 64, 85, 111, 147, 191, 245, 315, 405, 515, 652, 823, 1036, 1295, 1617, 2011, 2493, 3076, 3788, 4650, 5696, 6952, 8464, 10280, 12461, 15059, 18163, 21858, 26255, 31463, 37642, 44933, 53555, 63704, 75654, 89683, 106163, 125445, 148021

Offset: 0

Author

Gus Wiseman, Jun 12 2021

Keywords

• nonn

Comments

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).

Examples

			The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)    (3)    (4)      (5)      (6)        (7)          (8)
       (1,1)  (2,1)  (2,2)    (3,2)    (3,3)      (4,3)        (4,4)
                     (3,1)    (4,1)    (4,2)      (5,2)        (5,3)
                     (2,1,1)  (2,2,1)  (5,1)      (6,1)        (6,2)
                              (3,1,1)  (3,2,1)    (3,2,2)      (7,1)
                                       (4,1,1)    (3,3,1)      (3,3,2)
                                       (2,2,1,1)  (4,2,1)      (4,2,2)
                                                  (5,1,1)      (4,3,1)
                                                  (3,2,1,1)    (5,2,1)
                                                  (2,2,1,1,1)  (6,1,1)
                                                               (3,2,2,1)
                                                               (3,3,1,1)
                                                               (4,2,1,1)
                                                               (2,2,2,1,1)
                                                               (3,2,1,1,1)
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).
		

Links

• Joseph Likar, Table of n, a(n) for n = 0..1000

Crossrefs

The complement is counted by A344654.

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

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

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

A001250 counts wiggly permutations.

A003242 counts anti-run compositions.

A025047 counts wiggly compositions (ascend: A025048, descend: A025049).

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344604 counts wiggly compositions with twins.

A344605 counts wiggly patterns with twins.

A344606 counts wiggly permutations of prime indices with twins.

A344614 counts compositions with no consecutive strictly monotone triple.

A345164 counts wiggly permutations of prime indices.

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

A345192 counts non-wiggly compositions.

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

Programs

Mathematica

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

Formula

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

Extensions

a(26)-a(32) from Robert Price, Jun 22 2021

a(33) onwards from Joseph Likar, Sep 05 2023