A344654 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.

0, 0, 0, 1, 1, 2, 4, 5, 7, 11, 16, 20, 28, 37, 50, 65, 84, 106, 140, 175, 222, 277, 350, 432, 539, 663, 819, 999, 1225, 1489, 1816, 2192, 2653, 3191, 3846, 4603, 5516, 6578, 7852, 9327, 11083, 13120, 15532, 18328, 21620, 25430, 29904, 35071, 41110, 48080

Offset: 0

Author

Gus Wiseman, Jun 12 2021

Keywords

• nonn

Comments

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).

Examples

			The a(3) = 1 through a(9) = 11 partitions:
  (111)  (1111)  (2111)   (222)     (2221)     (2222)      (333)
                 (11111)  (3111)    (4111)     (5111)      (3222)
                          (21111)   (31111)    (41111)     (6111)
                          (111111)  (211111)   (221111)    (22221)
                                    (1111111)  (311111)    (51111)
                                               (2111111)   (321111)
                                               (11111111)  (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
                                                           (111111111)
		

Links

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

Crossrefs

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

The complement is counted by A344740.

The normal case starts 0, 0, 0, then becomes A345162, complement A345163.

Allowing twins (x,x) gives A345165, ranked by A345171.

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.

A345170 counts partitions with a wiggly permutation, ranked by A345172.

A345192 counts non-wiggly compositions.

Cf. A000041, A000070, A102726, A103919, A333489, A335126, A344607, A344615, A345166, A345168, A345169.

Programs

Mathematica

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

Extensions

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

a(33) onwards from Joseph Likar, Sep 06 2023