|
|
A216256
|
|
Minimum length of a longest unimodal subsequence of a permutation of n elements.
|
|
2
|
|
|
1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
a(n) is the value such that for any permutation P of n elements, P always contains a unimodal subsequence of length a(n), i.e., a sequence that is increasing, or decreasing, or increasing then decreasing.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = ceiling(sqrt(3*n - 3/4) - 1/2).
|
|
EXAMPLE
|
a(3) = 3 because all permutations of 3 elements are unimodal.
a(4) = 3 because there are permutations of 4 elements (e.g., 1423) that are not unimodal, but using the previous value we can always fix that by deleting one element.
|
|
MAPLE
|
|
|
MATHEMATICA
|
Table[Ceiling[Sqrt[3 n - 3/4] - 1/2], {n, 100}] (* Wesley Ivan Hurt, Oct 16 2015 *)
|
|
PROG
|
(C) unsigned int a(unsigned int n) { return ceil( sqrt((double) 3*n - 0.75) - 0.5); }
(PARI) a(n) = ceil(sqrt(3*n-3/4) - 1/2); \\ Michel Marcus, Apr 22 2014
(Magma) [Ceiling(Sqrt(3*n - 3/4) - 1/2) : n in [1..100]]; // Wesley Ivan Hurt, Oct 16 2015
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|