For each positive integer $n$, split the integers $1$ to $2n$ into two sets of $n$ elements each, and such that the products of the elements in each of these sets are as close as possible, say they differ by $a(n)$.


It can be checked that $a(1)=1$, $a(2)=2$, $a(3)=6$, $a(4)=18$, and $a(5)=30$.


Is this sequence strictly increasing?


What about if it is not required that the two sets contain the same number of elements?



Splitting the integers from $1$ to $2n$ into two sets with products as close as possible
Original question
-----------------



> 
> Is this sequence strictly increasing?
> 
> 
> 


No.



```
n  difference     smaller half
16 16753029012720 [3, 5, 6, 7, 9, 10, 11, 13, 15, 18, 19, 21, 25, 27, 29, 30]
17 10176199188480 [4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 18, 19, 21, 22, 27, 28, 33]

```


> 
> What about if it is not required that the two sets contain the same number of elements?
> 
> 
> 


The same counterexample applies.


Further questions arising
-------------------------


### Is the requirement for equally sizes parts at all relevant?


[Rob Pratt](https://mathoverflow.net/users/141766/robpratt) commented that for $n \le 10$ the optimal split without the requirement for equally sized parts is attainable with equally sized parts. This continues to hold up to $n = 30$:



```
1                                 1 [1]
2                                 2 [1, 4]
3                                 6 [2, 3, 4]
4                                18 [2, 3, 4, 8]
5                                30 [2, 3, 5, 7, 9]
6                               576 [2, 4, 5, 6, 9, 10]
7                               840 [3, 4, 5, 6, 7, 9, 13]
8                             24480 [2, 4, 6, 8, 9, 10, 11, 12]
9                             93696 [2, 4, 5, 7, 8, 10, 14, 15, 17]
10                           800640 [3, 4, 5, 7, 8, 10, 13, 14, 15, 17]
11                          7983360 [4, 5, 6, 7, 8, 9, 11, 12, 13, 17, 19]
12                         65318400 [4, 5, 6, 7, 8, 9, 12, 14, 15, 16, 17, 19]
13                       2286926400 [3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 18, 21, 26]
14                      13680979200 [3, 5, 6, 7, 9, 10, 11, 12, 15, 18, 20, 22, 23, 27]
15                     797369149440 [3, 5, 6, 8, 9, 10, 12, 13, 15, 17, 18, 20, 25, 26, 27]

18                  159943859712000 [1, 3, 5, 7, 8, 10, 11, 12, 13, 15, 16, 17, 20, 22, 24, 30, 31, 32, 33]
19                26453863460044800 [1, 3, 5, 6, 7, 8, 9, 12, 13, 14, 18, 19, 21, 24, 27, 28, 35, 36, 37, 38]
20               470500040794291200 [1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 18, 19, 23, 24, 26, 27, 29, 31, 36, 37, 39]
21             20720967220237197312 [1, 3, 4, 7, 8, 9, 12, 13, 14, 16, 17, 18, 23, 24, 26, 27, 28, 29, 32, 36, 39, 41]
22             61690805562507264000 [1, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 26, 28, 30, 31, 32, 34, 35, 42]
23           9203996481363478738944 [1, 3, 4, 6, 7, 9, 11, 12, 13, 17, 18, 19, 21, 26, 27, 29, 31, 33, 36, 37, 38, 41, 42, 43]
24         226577104515475594214400 [1, 3, 5, 6, 7, 9, 11, 12, 14, 17, 19, 21, 22, 23, 27, 28, 29, 33, 34, 35, 37, 38, 41, 42, 47]
25        4571875103611079835648000 [1, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 19, 20, 22, 24, 26, 27, 29, 32, 36, 37, 39, 41, 43, 47, 48]
26       20218804109333464320000000 [1, 3, 5, 7, 9, 10, 13, 14, 15, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 36, 39, 41, 43, 46, 47, 52]
27     3678271958960426245017600000 [1, 3, 5, 7, 10, 11, 13, 14, 15, 17, 18, 19, 20, 22, 23, 25, 26, 29, 30, 31, 34, 39, 41, 43, 46, 48, 53, 54]
28   217018448461953024000491520000 [2, 4, 6, 8, 9, 11, 12, 15, 16, 18, 23, 24, 26, 27, 30, 32, 33, 36, 37, 38, 39, 43, 44, 45, 46, 48, 54, 55]
29 18646773190859199121234329600000 [1, 3, 5, 7, 10, 11, 13, 14, 15, 19, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 35, 36, 38, 39, 42, 43, 48, 49, 52, 57]
30 90984319783113193178231808000000 [1, 3, 5, 7, 9, 10, 11, 14, 15, 18, 19, 20, 21, 22, 23, 27, 28, 31, 33, 35, 37, 41, 42, 43, 44, 46, 47, 49, 54, 55, 56]

```

It gets quite tight, though: for $28$, once multiplicities of other primes are taken into account there's very almost a shortfall of powers of three. I need to improve my search code to take this much further.


Asymptotics
-----------


[Seva](https://mathoverflow.net/users/9924/seva) and [David E Speyer](https://mathoverflow.net/users/297/david-e-speyer) address the question of asymptotics. The observation that for known values the difference rarely exceeds $n!$ ($n=29$ is the only exception, with a difference of $\sim 2.1 n!$) puts it on the level of Seva's abc-conjecture-conditional lower bound, and beats David's guess that the asymptotic is $\sqrt{(2n)!} e^{-o(n)}$. I have been no more successful than David in finding a strategy which beats $\sqrt{(2n)!}n^{-C}$, but I think there's an argument in the spirit of Erdős for why it's reasonable to hope that the asymptotic is $\Theta(n!)$ (or $\sqrt{(2n)!} e^{-cn}$).


If we want a difference less than $n!$ then to first order we want the smaller half's product to be $\ge \sqrt{(2n)!} - \tfrac12 n!$. The products of half of the terms pair up, and the smaller half ranges from $n!$ to $\sqrt{(2n)!}$. There are $\frac{(2n)!}{2n!^2}$ smaller halves. If their logarithms are distributed evenly in the given range then asymptotically we expect there to be far more than one above the desired threshold. So the question is how unevenly distributed the logarithms are. Calculating that is outside my skillset, but I can build an empirical table of the number of products which fall within the required range for small $n$. The headers abbreviate asymmetric (i.e. without the requirement for parts of equal size), heuristic, and symmetric (with the requirement). I've generally rounded the heuristic down unless the fractional part is above 0.9. Note that these calculations were done on the basis of exact thresholds, not the first order approximation.



```
 n   asym (heur)      sym (heur)

 8     14 (9)          14 (6)
 9      2 (16)          2 (10)
10     21 (28)         21 (17)
11    312 (51)        279 (31)
12    351 (93)        319 (55)
13     64 (169)        59 (99)
14   1849 (312)      1637 (180)
15     42 (577)        39 (330)
16   2777 (1073)     2383 (606)
17    292 (2007)      258 (1119)
18  10725 (3768)     9164 (2077)
19   3829 (7099)     3215 (3870)
20    530 (13421)     459 (7237)

```

