A332726 Number of compositions of n whose run-lengths are unimodal.

1, 1, 2, 4, 8, 16, 31, 61, 120, 228, 438, 836, 1580, 2976, 5596, 10440, 19444, 36099, 66784, 123215, 226846, 416502, 763255, 1395952, 2548444, 4644578, 8452200, 15358445, 27871024, 50514295, 91446810, 165365589, 298730375, 539127705, 972099072, 1751284617, 3152475368

Offset: 0

Author

Gus Wiseman, Feb 29 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 only composition of 6 whose run-lengths are not unimodal is (1,1,2,1,1).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• Eric Weisstein's World of Mathematics, Unimodal Sequence.

Crossrefs

Looking at the composition itself (not run-lengths) gives A001523.

The case of partitions is A332280, with complement counted by A332281.

The complement is counted by A332727.

Unimodal compositions are A001523.

Unimodal normal sequences appear to be A007052.

Non-unimodal compositions are A115981.

Compositions with normal run-lengths are A329766.

Numbers whose prime signature is not unimodal are A332282.

Partitions whose 0-appended first differences are unimodal are A332283, with complement A332284, with Heinz numbers A332287.

Compositions whose negated run-lengths are unimodal are A332578.

Compositions whose negated run-lengths are not unimodal are A332669.

Compositions whose run-lengths are weakly increasing are A332836.

Cf. A072706, A100883, A181819, A227038, A328509, A329744, A329746, A332642, A332670, A332741, A332833, A332835.

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[Length/@Split[#]]&]],{n,0,10}]

PARI

step(M, m)={my(n=matsize(M)[1]); for(p=m+1, n, my(v=vector((p-1)\m, i, M[p-i*m,i]), s=vecsum(v)); M[p,]+=vector(#M,i,s-if(i<=#v, v[i]))); M}
desc(M, m)={my(n=matsize(M)[1]); while(m>1, m--; M=step(M,m)); vector(n, i, vecsum(M[i,]))/(#M-1)}
seq(n)={my(M=matrix(n+1, n+1, i, j, i==1), S=M[,1]~); for(m=1, n, my(D=M); M=step(M, m); D=(M-D)[m+1..n+1,1..n-m+2]; S+=concat(vector(m), desc(D,m))); S} \\ Andrew Howroyd, Dec 31 2020

Formula

a(n) + A332727(n) = 2^(n - 1).

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020