A352813 Minimum difference |product(A) - product(B)| where A and B are a partition of {1,2,3,...,2*n} and |A| = |B| = n.

0, 1, 2, 6, 18, 30, 576, 840, 24480, 93696, 800640, 7983360, 65318400, 2286926400, 13680979200, 797369149440, 16753029012720, 10176199188480, 159943859712000, 26453863460044800, 470500040794291200, 20720967220237197312, 61690805562507264000

Offset: 0

Author

Peter J. Taylor, Apr 04 2022

Keywords

• nonn

Comments

a(n) >= A038667(2*n).

Conjecture: a(n) = A038667(2*n) for all n. It is verified for n<=70. - Max Alekseyev, Jun 18 2022

Bernardo Recamán Santos proposes that this should be called Luciana's sequence for the student whose question prompted its investigation. (See MathOverflow link below.)

Examples

			For n = 4, the partition A = {1,5,6,7} and B = {2,3,4,8} is optimal, giving difference 1*5*6*7 - 2*3*4*8 = 18.
_Rob Pratt_ computed the optimal solutions for n <= 10:
[ n]    a(n)                   partitions of 2n
------------------------------------------------------------------
[ 1]       1                         2 | 1
[ 2]       2                       2,3 | 1,4
[ 3]       6                     1,5,6 | 2,3,4
[ 4]      18                   1,5,6,7 | 2,3,4,8
[ 5]      30                2,3,4,8,10 | 1,5,6,7,9
[ 6]     576              1,4,7,8,9,11 | 2,3,5,6,10,12
[ 7]     840           2,4,5,6,8,11,14 | 1,3,7,9,10,12,13
[ 8]   24480        1,5,6,7,8,13,14,15 | 2,3,4,9,10,11,12,16
[ 9]   93696     2,3,6,8,9,11,12,13,18 | 1,4,5,7,10,14,15,16,17
[10]  800640  2,3,4,8,9,11,12,18,19,20 | 1,5,6,7,10,13,14,15,16,17
		

Links

• Max Alekseyev, Table of n, a(n) for n = 0..70
• Gordon Hamilton, Thirsty Fractions, MathPickle, 2013, for elementary and middle school teachers.
• MathOverflow discussion Splitting the integers from 1 to 2n into two sets with products as close as possible.

Crossrefs

Cf. A061057, A038667.

Programs

Python

from math import prod, factorial
from itertools import combinations
def A352813(n):
    m = factorial(2*n)
    return 0 if n == 0 else min(abs((p:=prod(d))-m//p) for d in combinations(range(2,2*n+1),n-1)) # Chai Wah Wu, Apr 06 2022

Sage

def A352813(n):
    return min(abs(prod(A)-prod(B)) for (A,B) in SetPartitions((1..2*n), [n,n]))
[A352813(n) for n in (1..10)] # Freddy Barrera, Apr 05 2022