A332669 Number of compositions of n whose negation is not unimodal.

0, 0, 0, 0, 1, 3, 11, 28, 71, 165, 372, 807, 1725, 3611, 7481, 15345, 31274, 63392, 128040, 257865, 518318, 1040277, 2085714, 4178596, 8367205, 16748151, 33515214, 67056139, 134147231, 268341515, 536746350, 1073577185, 2147266984, 4294683056, 8589563136, 17179385180

Offset: 0

Author

Gus Wiseman, Feb 28 2020

Keywords

• nonn

Comments

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

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

Examples

			The a(4) = 1 through a(6) = 11 compositions:
  (121)  (131)   (132)
         (1121)  (141)
         (1211)  (231)
                 (1131)
                 (1212)
                 (1221)
                 (1311)
                 (2121)
                 (11121)
                 (11211)
                 (12111)
		

Links

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

Crossrefs

The strict case is A072707.

The complement is counted by A332578.

The version for run-lengths of partitions is A332639.

The version for unsorted prime signature is A332642.

The version for 0-appended first-differences of partitions is A332744.

The case that is not unimodal either is A332870.

Unimodal compositions are A001523.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Non-unimodal normal sequences are A328509.

Numbers whose unsorted prime signature is not unimodal are A332282.

A triangle for compositions with unimodal negation is A332670.

Cf. A007052, A072706, A227038, A329398, A332281, A332284, A332638, A332728, A332742, A332832.

Programs

Mathematica

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[-#]&]],{n,0,10}]

Formula

a(n) + A332578(n) = 2^(n - 1) for n > 0.

Extensions

Terms a(21) and beyond from Andrew Howroyd, Mar 01 2020