A115981 The number of compositions of n which cannot be viewed as stacks.

0, 0, 0, 0, 0, 1, 5, 17, 49, 126, 303, 694, 1536, 3312, 7009, 14619, 30164, 61732, 125568, 254246, 513048, 1032696, 2074875, 4163256, 8345605, 16717996, 33473334, 66998380, 134067959, 268233386, 536599508, 1073378850, 2147000209

Offset

0,7

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. - Gus Wiseman, Mar 05 2020

Links

Table of n, a(n) for n=0..32.

Eric Weisstein's World of Mathematics, Unimodal Sequence

Formula

a(n) = A011782(n) - A001523(n).

Example

a(5) = 1 counting {212}.
a(6) = 5 counting {1212, 2112,2121,213,312}.
a(7) = 17 counting {11212, 12112,12121, 21211, 21121, 21112, 2122, 2212, 2113, 3112, 2131, 3121, 1213, 1312, 412, 214, 313}.
a(8) = 49 = 128 - 79.
a(9) = 126 = 256 - 130.

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}] (* Gus Wiseman, Mar 05 2020 *)

Crossrefs

Cf. A011782, A115982.

The complement is counted by A001523.

The strict case is A072707.

The case covering an initial interval is A332743.

The version whose negation is not unimodal either is A332870.

Non-unimodal permutations are A059204.

Non-unimodal normal sequences are A328509.

Partitions with non-unimodal run-lengths are A332281.

Numbers whose prime signature is not unimodal are A332282.

Partitions whose 0-appended first differences are not unimodal are A332284.

Non-unimodal permutations of the prime indices of n are A332671.

Cf. A007052, A072704, A227038, A329398, A332280, A332283, A332672, A332578, A332669, A332834.

Sequence in context: A268783 A273384 A006457 * A083091 A176953 A082753

Adjacent sequences: A115978 A115979 A115980 * A115982 A115983 A115984

Keyword

easy,nonn

Author

Alford Arnold, Feb 12 2006

Extensions

More terms from Brian Kuehn (brk158(AT)psu.edu), Apr 20 2006

a(25) corrected by Georg Fischer, Jun 29 2021

Status

approved