A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

1, 1, 2, 4, 8, 16, 29, 56, 101, 181, 327, 583, 1023, 1820, 3207, 5631, 9905, 17394, 30489, 53481, 93725, 164169, 287606, 503672, 881834, 1544018, 2703161, 4731860, 8283291, 14499392, 25379278, 44422866, 77754798, 136093756, 238204369, 416923752, 729728031

Offset: 0

Author

Gus Wiseman, Feb 29 2020

Keywords

• nonn

Comments

A composition of n is a finite sequence of positive integers summing to n.

Examples

			The a(6) = 29 compositions:
  (6)    (141)  (213)   (1113)  (21111)
  (51)   (114)  (132)   (222)   (12111)
  (15)   (33)   (123)   (2211)  (11121)
  (42)   (321)  (3111)  (2121)  (11112)
  (24)   (312)  (1311)  (1212)  (111111)
  (411)  (231)  (1131)  (1122)
Missing are: (2112), (1221), (11211).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Unimodal Sequence

Crossrefs

The version for the compositions themselves (not run-lengths) is A329398.

Compositions with equal run-lengths are A329738.

The case of partitions is A332745.

The version for unsorted prime signature is the complement of A332831.

The complement is counted by A332833.

Unimodal compositions are A001523.

Partitions with weakly decreasing run-lengths are A100882.

Partitions with weakly increasing run-lengths are A100883.

Compositions that are not unimodal are A115981.

Compositions whose negation is unimodal are A332578.

Compositions whose run-lengths are unimodal are A332726.

Neither weakly increasing nor weakly decreasing compositions are A332834.

Compositions with weakly increasing run-lengths are A332836.

Compositions that are neither unimodal nor is their negation are A332870.

Cf. A001462, A181819, A329744, A329766, A332273, A332640, A332641, A332746.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,20}]

Formula

a(n) = 2 * A332836(n) - A329738(n).

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020