I'm curious about how these would rank out mathematically, too. I have most of these figured out using sets, but the straights, and flushes are difficult for me to figure out, especially when you consider 5, 6, and 7 card versions. For just sets (pairs P, triples T, and quads Q) the ranks are as follows:
1P (63258624) = 13x6x12C5x(4^5)
2P (29652480) = 13C2x(6^2)x11C3x(4^3)
0P (28114944) = 13C7x(4^7)
1T (6589440) = 13x4x12C4x(4^4)
1T1P (3294720) =13x4x12x6x11C2x(4^2)
3P (2471040) = 13C3x(6^3)x10x4
1Q (183040) = 13x12C3x(4^3)
1T2P (123552) = 13x4x12C2x(6^2)
2T (54912) = 13C2x(4^2)x11x4
1Q1P (41184) = 13x12x6x11x4
1Q1T (624) = 13x12x4
The 1T2P, 1Q1P and 1Q1T hands all lack names, but, I think I'd call them Duplex, Mansion, and Castle, respectively.
Now, what I do know is which hands the Straights, Flushes, and Straight Flushes (here I'll just say SF) are special cases of:
5SF: 0P, 1P, 2P, 1T
6SF: 0P, 1P
Ex. 5-Straight, 2 Pair: A, A, K, K, Q, J, 10.
5-Straight, 1 Triple: A, A, A, K, Q, J, 10.
5-Straight, 1 Pair: A, A, K, Q, J, 10, 2.
5-Straight, 1 Pair: A, K, Q, J, 10, 2, 2.
5-Straight, No Pair: A, K, Q, J, 10, 3, 2.
6-Straight, 1 Pair: A, A, K, Q, J, 10, 9.
6-Straight, No Pair: A, K, Q, J, 10, 9, 2.
7-Straight: A, K, Q, J, 10, 9, 8.
Now, this creates three different sets of ranks: one where SF hands can have Pairs and Triples using the same cards, one where only 5SF and a Pair is a unique rank, and one where 5SF with a Pair is just a 5SF with two kickers at the same rank. Also possible, and arguably easier to remember is for 7 card hands to play as 7 card hands, meaning all 7 cards must be the same Suit for a Flush, or in a 7 card sequence for a Straight, or, ignore some of the actual ranking theory and make 5,6,7 card SF hands all rank the same as 5SF hands but the 6 and 7 card variants are higher in their respective ranks. These create 5 alternative ranking models, 3 have additional ranks and two do not.
The 3 with additional ranks have an additional 21, 9, and 6 ranks respectively.
The other two have no additional ranks. The 5,6,7 SF are all the same rank form is closer to regular 7 card games using the best 5 card hand principle, while the 7SF only form is closer to actually playing Poker with a 7 card hand (and arguably easier to calculate probabilities of)
The probabilities for the 7SF only variant is easy enough to calculate
Straight (131040) = 8x(4^7) - 32, between 1Q and 1T2P.
Flush (6832) = 13C7x4 - 32, between 1Q1P and 1Q1T.
Straight Flush (32) = 8x4, above 1Q1T.
Using regular best 5 card hand rules, pulled straight from Wikipedia, we have
Straight = 6180020, between 1T and 1T1P.
Flush = 4047644, between 1T and 1T1P.
Straight Flush = 41584, Above 2T, below 1Q1P.
Mind, I did not subtract any of these from purely Pair, Triple, and Quad based ranks, so there is a double count here with these not pulling their possibilities from 0P, 1P, 2P, and 1T hands. To do this, I would have to calculate with all 21 additional possibilities to find where I need to subtract how much. Advantage to this more exhaustive method: I can then produce rankings for all 5 possible systems (duplicates allowed, no duplicates, no duplicates but 5SF pair, best 5, only 7).