i7 3820 is a sidegrade, but fully capable of matching whatever your 2600K is able to do, and would be the path I would take if I were in your shoes
The advantages would be in the platform:
1. you would have more PCI-e lanes for adding expansion cards
2. more RAM slots if you choose a motherboard with more than 4 (most have 8, although the two less-expensive ROG boards only have 4)
3. best chance for upgrading CPU in the future
the last point is the most valid, especially if you go with the i7 3820. While there's no guarantee that IB-E will be compatible with current X79 motherboards (although all current indications suggest they will be), if you go with the i7 3820 you will still be able to drop in a 6 core SB-E to upgrade the 3820. There would be no such option for s1155 as it is going to be replaced by s1150 with Haswell, and also no guarantee that the Haswell platform will offer an upgrade in CPU performance for those who already have a high end intel quad.
Really, the primary reason to go LGA 2011 for you would be to give you a platform ready for being adapted to the future, while the primary reason to go for 3770K would be that its likely a slightly cheaper alternative*, and really neither platform will offer much of an upgrade
*assuming no Microcenter option and keeping similar motherboards, the ASUS Rampage IV Gene is $90 more than the Maximus V Gene, although 3820 is $30 less than the 3770, for a net $60 more to go LGA 2011, although who knows how prices with change by the time you will be ready to buy
What is your current cooler? If you need a new bracket for your cooler, it might be worth looking at Asrock boards as they have multiple mounting holes so not needing a new bracket.
As to the 3820 vs 3770K, the first is not really overclockable, and it is about on par with a 2500 IIRC, so it is definitly slower than a 3770K.
6 sticks can be used in the s2011 board, but for full advantage you will need a s2011 board with 8 slots (some come with only 4 for better ram overclocking) and still have to find 2 more matching ram sticks.
re upgrade path, neither have a very good upgrade path currently as both are about end of their life. IB-E will be a IvyBridge for s2011 so it is expected the Ivybridge replacement cpu (due first) will be faster for home users. Added to that, if you check the price of current SB-E cpus, that is the expected price points intel will place the IB-E cpus at, so if those higher end SB-E's scare you, avoid the s2011 path.
there's quite a bit of misinformation here
SB-E has BCLK straps that allow for BCLK overclocking, and thus while the i7 3820 does not have a fully unlocked multiplier (of which it still allows some play because of turbo), it can still overclock to the same speeds as other Sandy/Ivy CPUs. Heck, its stock multiplier (36) plus the first step up in BCLK (133 up from 100) yields an 'easy' 4.5GHz, with plenty of room to find combinations for other clockrates higher and lower, its just a teeny bit harder than only changing the multiplier.
as far as RAM goes, you don't necessarily have to have matching sticks as long as you run them at the lowest common denominator, heck, many X79 boards allow you to supply each channel with varying voltages to help with that syncro
and as far as upgrade path, if anything s1155 is currently dead in the water with no real upgrade opportunity in sight, whereas s2011 and X79 currently looks to be compatible with IvyBridge-E which will likely offer 8+ cores as well as the likelihood of solder TIM and thus superior overclocking to 1155 Ivy. On top of that, while an i7 3820 would be a sidegrade, the platform would still have the potential to upgrade, even if its not to IB-E there's still 6 core SB-E like the 3930K, and very much an option to buy one used off of someone (like myself) for a fair price once they make the upgrade to IvyBridge-E. On the flip side there would be no such option with a 3770K rig, as Haswell is going to be a new socket and likely won't offer much CPU performance improvement anyway now that intel is making a pretty significant shift to improving iGPU.