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I did, however, get two Emperor's Crowns on a run the other night.
Omega Ferromin is a serious pain to farm. Literally the only purpose of rank 4-5 items is to farm more rank 4-5 items, so x3 Omega Ferromin for Invincible Armor is a bit bonkers. Doesn't help that both Louie and Arma, the only characters who can wear Invincible Armor, are both heavily outclassed for farming purposes by Griff and Charme. (Arma might have Rocket Boots, but give Charme some good defensive equipment and like 30 mont blancs and she can spam Flame Charge from start to finish of a thorough 10 floor run without running out of inventory space)
Thanks again for clearing up my doubts regarding the last couple of items.
Wow, could you actually describe what your effort was? Did you just save/load multiple times, or it didn't help? Did you keep leveling her up by buying at near pin prices? I understand it's been two years already, but still. Nobody else on the internet says what they did to get the carpet, or whether they got it at all.
Yes you must do something special before you can get the Sea Carpet. Euria will only attempt to sell you the Sea Carpet if you have both the Ocean Wallpaper and Seabed Floor decorations active in the store. Once this happens and she visits, she has a 1 in 12 chance of selling it to you.
The Emperor's Crown is simply rare. There's a 1 in 4480 chance at best from getting it from a chest in the Crystal Nightmare. The left chest on floor 30/60 (basically if you started from floor 1 or did the boss rush) has a 1 in 224 chance of giving it to you instead.
The right chest has a 1 in 946 chance of giving you the crown.
The middle chest has a 1 in 61.632~ chance of giving you the crown, so long as you started from the boss rush, or at least started from floor 11. If you started from floor 21, the chance falls to 1 in 1892 instead.
Overall you have around a 1 in 46 chance of finding one at the end of the dungeon if you start from floor 1 or do the boss rush.
As for the Omega Ferromin... the rarity is unfortunately glitched. A mistake in the RNG code causes the item to be rarer than intended. It's supposed to have the same rarity as the Emperor's Crown in some cases, but that unfortunately isn't the case.
At best you have a 1 in 1763 chance of obtaining one from the Crystal Nightmare. You can also find it in Lapis Ruins but the chance of getting one from a chest is 1 in 8780 at best.
The left chest at the end floor has a 1 in 448 chance of giving you one. The right chest has about a 1 in 631 chance of giving it to you. And the middle chest has a 1 in 1892 chance of giving it to you if you started from the boss rush or from floor 11. If you started from floor 21, the chance is... boosted to 1 in 1261 instead. This part isn't a mistake, it's just how the end rewards of the Crystal Nightmare are coded.
The overall chance of finding Omega Ferromin at the end of the dungeon is 1 in 230. If the RNG worked as intended, the chance would be 1 in 137 instead, but alas.