The On-Line Encyclopedia of Integer Sequences (OEIS)

A344740 revision #16

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

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OFFSET

0,3

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).

LINKS

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

FORMULA

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

EXAMPLE

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).

MATHEMATICA

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

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.

Sequence in context: A239945 A363740 A238875 * A241391 A241736 A034398

Adjacent sequences: A344737 A344738 A344739 * A344741 A344742 A344743

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 12 2021

EXTENSIONS

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

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

STATUS

proposed