

A115981


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


98



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
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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.
Nonunimodal permutations are A059204.
Nonunimodal normal sequences are A328509.
Partitions with nonunimodal runlengths are A332281.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0appended first differences are not unimodal are A332284.
Nonunimodal 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



