A332746 Number of integer partitions of n such that either the run-lengths or the negated run-lengths are unimodal.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 134, 174, 227, 291, 373, 473, 598, 748, 936, 1163, 1437, 1771, 2170, 2651, 3226, 3916, 4727, 5702, 6846, 8205, 9793, 11681, 13866, 16462, 19452, 22976, 27041, 31820, 37276, 43693, 51023, 59559, 69309, 80664

Offset

0,3

Comments

First differs from A000041 at a(14) = 134, A000041(14) = 135.

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Links

Table of n, a(n) for n=0..47.

MathWorld, Unimodal Sequence

Example

The only partition not counted under a(14) = 134 is (4,3,3,2,1,1), whose run-lengths (1,2,1,2) are neither unimodal nor is their negation.

Mathematica

unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n], unimodQ[Length/@Split[#]]||unimodQ[-Length/@Split[#]]&]], {n, 0, 30}]

Crossrefs

Looking only at the original run-lengths gives A332281.

Looking only at the negated run-lengths gives A332639.

The complement is counted by A332640.

The Heinz numbers of partitions not in this class are A332643.

Unimodal compositions are A001523.

Partitions with unimodal run-lengths are A332280.

Compositions whose negation is unimodal are A332578.

Partitions whose negated run-lengths are unimodal are A332638.

Run-lengths are neither weakly increasing nor weakly decreasing: A332641.

Run-lengths and negated run-lengths are both unimodal: A332745.

Cf. A007052, A025065, A100883, A115981, A181819, A332283, A332577, A332578, A332642, A332669, A332726, A332831.

Sequence in context: A008635 A008641 A341714 * A242698 A364793 A330641

Adjacent sequences: A332743 A332744 A332745 * A332747 A332748 A332749

Keyword

nonn

Author

Gus Wiseman, Feb 27 2020

Status

approved