The Triton poker series has popularized short deck hold-em. I am sure that everyone knows by now that Short Deck is played with a 36-card deck since all the 2's, 3's, 4's, and 5's have been removed (of course, this is why Short Deck is also called Six-Plus poker).

It is a wildly fast-moving game with frequent large pots and tons of action. Pre-flop equities run much tighter than in Long Deck (regular) hold-em. Similar to PLO, the in-hand swings are wild it being common for each card (flop, turn, river) to greatly change who is ahead in the hand.

Since there are fewer cards in each suit, and five still being required for a flush, flushes in Short Deck are less common than in Long Deck. Also, since there are fewer ranks in each suit, full houses are more common in Short Deck than in Long Deck. Accordingly, in Short Deck a flush beats a full house.

Here I want to look into the differential frequency of a flush in Short Deck vs Long Deck. In particular, I will look at how often a flush is made by at least one player at a six-max table if each hand goes to showdown on each deal. Also, to keep the comparison "apples-to-apples", I will ignore full houses in both cases (obviously a full house beats a flush in Long Deck). This initial foray is intended to be pure math based.


Long Deck (regular 52-card hold-em):

I will take the total number of deals of 6-Max Long Deck to be:
= C(52,5)*C(47,12)*(1*3*5*7*9*11)
= 1,411,633,731,355,657,009,200

Now let's go through all the ways that one or more players can make a flush.

Case 1: 5 board cards of same suit
= C(4,1)*C(13,5)*C(47,12)*(1*3*5*7*9*11)
= 2,796,153,249,383,954,460

Case 2: Exactly 4 board cards of same suit
In this case, of course, anybody with at least one hole card of that suit will have a flush. To calculate this, I will subtract the total from the case in which nobody has a hole card of that suit.
= C(4,1)*C(13,4)*C(39,1)*[C(47,12)-C(38,12)]*(1*3*5*7*9*11)
= 57,444,115,870,926,684,900

Case 3: Exactly 3 board cards of same suit

Case 3A: Exactly 5 players make flush
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)*C(6,2)*C(4,2)*C(2,2)/5!]*C(37,2)
= 533,519,466,480

Case 3B: Exactly 4 players make flush

Case 3B1: 2 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)*C(6,2)*C(4,2)/4!]*C(37,2)*C(2,2)*2
= 5,335,194,664,800

Case 3B2: 1 other suit card in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)*C(6,2)*C(4,2)/4!]*C(37,3)*C(2,1)*3
= 186,731,813,268,000

Case 3B3: 0 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)*C(6,2)*C(4,2)/4!]*C(37,4)*C(2,0)*3
= 793,610,206,389,000

Case 3C: Exactly 3 players make flush

Case 3C1: 3 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)*C(6,2)/3!]*C(37,3)*C(4,3)*3!
= 497,951,502,048,000

Case 3C2: 2 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)*C(6,2)/3!]*C(37,4)*C(4,2)*(C(4,2)*2)
= 12,697,763,302,224,000

Case 3C3: 1 other suit card in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)*C(6,2)/3!]*C(37,5)*C(4,1)*(1*3*5)
= 69,837,698,162,232,000

Case 3C4: 0 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)*C(6,2)/3!]*C(37,6)*C(4,0)*(1*3*5)
= 93,116,930,882,976,000

Case 3D: Exactly 2 players make flush

Case 3D1: 4 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)/2!]*C(37,4)*C(6,4)*4!
= 12,697,763,302,224,000

Case 3D2: 3 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)/2!]*C(37,5)*C(6,3)*(C(5,2)*3!)
= 279,350,792,648,928,000

Case 3D3: 2 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)/2!]*C(37,6)*C(6,2)*(C(6,2)*2*3)
= 1,676,104,755,893,568,000

Case 3D4: 1 other suit card in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)/2!]*C(37,7)*C(6,1)*(1*3*5*7)
= 3,463,949,828,846,707,200

Case 3D5: 0 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*[C(10,2)*C(8,2)/2!]*C(37,8)*C(6,0)*(1*3*5*7)
= 2,164,968,643,029,192,000

Case 3E: Exactly 1 player makes flush

Case 3E1: 5 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*C(10,2)*C(37,5)*C(8,5)*5!
= 111,740,317,059,571,200

Case 3E2: 4 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*C(10,2)*C(37,6)*C(8,4)*(C(6 ,2)*4!)
= 2,234,806,341,191,424,000

Case 3E3: 3 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*C(10,2)*C(37,7)*C(8,3)*(C(7 ,4)*3*3!)
= 13,855,799,315,386,868,288

Case 3E4: 2 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*C(10,2)*C(37,8)*C(8,2)*(C(8 ,6)*(1*3*5)*2)
= 34,639,498,288,467,072,000

Case 3E5: 1 other suit card in other hole cards
= C(4,1)*C(13,3)*C(39,2)*C(10,2)*C(37,9)*C(8,1)*(1*3 *5*7*9)
= 35,876,623,227,340,896,000

Case 3E6: 0 other suit cards in other hole cards
= C(4,1)*C(13,3)*C(39,2)*C(10,2)*C(37,10)*C(8,0)*(1* 3*5*7*9)
= 12,556,818,129,569,313,600

GRAND TOTAL = 167,289,763,077,629,671,928

PERCENT = 11.851%


Short Deck (36-card Six Plus hold-em):

Here are the parallel calculations for Short Deck. I will take the total number of deals of 6-Max Short Deck to be:
= C(36,5)*C(31,12)*(1*3*5*7*9*11)
= 553,027,606,647,516,000

Now let's go through all the ways that one or more players can make a flush.

Case 1: 5 board cards of same suit
= C(4,1)*C(9,5)*C(31,12)*(1*3*5*7*9*11)
= 739,341,720,117,000

Case 2: Exactly 4 board cards of same suit
= C(4,1)*C(9,4)*C(27,1)*[C(31,12)-C(26,12)]*(1*3*5*7*9*11)
= 18,596,094,944,427,000

Case 3: Exactly 3 board cards of same suit

Case 3A: Exactly 3 players make flush
= C(4,1)*C(9,3)*C(27,2)*[C(6,2)*C(4,2)*C(2,2)/3!]*C(25,6)
= 313,296,984,000

Case 3B: Exactly 2 players make flush

Case 3B1: 2 other suit cards in other hole cards
= C(4,1)*C(9,3)*C(27,2)*[C(6,2)*C(4,2)/2]*C(25,6)*C(2,2)*(C(6,4)*3*2)
= 84,590,185,680,000

Case 3B2: 1 other suit card in other hole cards
= C(4,1)*C(9,3)*C(27,2)*[C(6,2)*C(4,2)/2]*C(25,7)*C(2,1)*(1*3*5*7)
= 535,737,842,640,000

Case 3B3: 0 other suit cards in other hole cards
= C(4,1)*C(9,3)*C(27,2)*[C(6,2)*C(4,2)/2]*C(25,8)*C(2,0)*(1*3*5*7)
= 602,705,072,970,000

Case 3C: Exactly 1 player makes flush

Case 3C1: 4 other suit cards in other hole cards
= C(4,1)*C(9,3)*C(27,2)*C(6,2)*C(25,6)*C(4,4)*(C(6,2 )*4!)
= 112,786,914,240,000

Case 3C2: 3 other suit cards in other hole cards
= C(4,1)*C(9,3)*C(27,2)*C(6,2)*C(25,7)*C(4,3)*(C(7,4 )*3*3!)
= 2,142,951,370,560,000

Case 3C3: 2 other suit cards in other hole cards
= C(4,1)*C(9,3)*C(27,2)*C(6,2)*C(25,8)*C(4,2)*(C(8,6 )*(1*3*5)*2)
= 9,643,281,167,520,000

Case 3C4: 1 other suit card in other hole cards
= C(4,1)*C(9,3)*C(27,2)*C(6,2)*C(25,9)*C(4,1)*(1*3*5 *7*9)
= 13,661,314,987,320,000

Case 3C5: 0 other suit cards in other hole cards
= C(4,1)*C(9,3)*C(27,2)*C(6,2)*C(25,10)*C(4,0)*(1*3* 5*7*9)
= 5,464,525,994,928,000

GRAND TOTAL = 51,583,643,497,386,000

PERCENT = 9.327%


Summary

We see that flushes are less common in 6-max Short Deck than in 6-max Long Deck (9.327% vs 11.851%) when all hands go to showdown on all deals.

I hope this is somewhat illuminating and prompts other analyses of Short Deck, which may be the new poker frontier?

Comments welcome.