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A332774 Given n line segments, the k-th of which is drawn from (k,0) to (x_k,1) where {x_1,x_2,...,x_n} is a permutation of {1,2,...,n}, a(n) is the maximum number of distinct points at which line segments intersect. 1
0, 1, 2, 5, 8, 13, 17, 23, 30, 39, 47, 57, 67, 79, 90, 103 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

There is a bijection between points with y=0 and y=1.

a(n) <= n(n-1)/2.

a(n) >= floor(n^2/4), by considering the n-permutation sending i to (floor(n/2)+i) mod n. - Boon Suan Ho, Sep 07 2022

LINKS

Table of n, a(n) for n=1..16.

Arvin Ding, Java program

EXAMPLE

For n=3, draw a line segment from (0,0) to (1,1), from (1,0) to (2,1), and from (2,0) to (0,1). This would correspond to 2 distinct intersections; namely, these are at (5/3,2/3) and (7/3,1/3).

This case corresponds to the permutation where {x_1,x_2,x_3} is {2,3,1}.

For the other 5 permutations, there are at most 2 distinct intersections. Because of this a(3)=2.

This table displays n, a(n), and the lexicographically earliest permutation for the first 13 positive n:

.

n a(n) lexicographically earliest permutation

-- ---- ---------------------------------------

1 0 { 1}

2 1 { 2, 1}

3 2 { 2, 3, 1}

4 5 { 3, 4, 2, 1}

5 8 { 3, 5, 4, 2, 1}

6 13 { 5, 6, 4, 3, 1, 2}

7 17 { 4, 7, 6, 3, 5, 2, 1}

8 23 { 6, 7, 8, 5, 3, 4, 1, 2}

9 30 { 7, 9, 4, 8, 6, 3, 5, 2, 1}

10 39 { 9, 10, 7, 8, 4, 3, 6, 5, 2, 1}

11 47 { 9, 8, 11, 10, 7, 6, 5, 3, 4, 1, 2}

12 57 { 9, 12, 11, 8, 7, 10, 4, 3, 6, 5, 2, 1}

13 67 {10, 13, 12, 9, 8, 11, 5, 4, 7, 6, 3, 2, 1}

PROG

(Java) See Ding link

CROSSREFS

Sequence in context: A178752 A225255 A076145 * A256829 A342431 A049617

Adjacent sequences: A332771 A332772 A332773 * A332775 A332776 A332777

KEYWORD

nonn,more,hard

AUTHOR

Arvin Ding, Feb 23 2020

EXTENSIONS

a(13) from Giovanni Resta, Feb 23 2020

Lexicographic earliest permutation corrected by Alexander Yan, Apr 06 2022

a(14)-a(16) from Misha Lavrov, Sep 07 2022

STATUS

approved

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Last modified February 2 09:15 EST 2023. Contains 360004 sequences. (Running on oeis4.)