Re: unique showdown probabilities for two preflop hand matchups
 

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However, while your way of organizing the permutations is nice on paper, it looks problematic from an algorithmic perspective.

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I still don't see what difficulties there are and what alternative you offer.

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For example, given the matchup:

A1K1 vs. Q1J1

the "switch two suits" permutation doesn't apply. There aren't two suits to switch.

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There are always 4 suits. You can always apply all 24 permutations of the 4 suits.

[img]/images/graemlins/spade.gif[/img] <-> [img]/images/graemlins/heart.gif[/img] applied to A[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/spade.gif[/img] vs. Q[img]/images/graemlins/spade.gif[/img] J[img]/images/graemlins/spade.gif[/img] results in A[img]/images/graemlins/heart.gif[/img] K[img]/images/graemlins/heart.gif[/img] vs. Q[img]/images/graemlins/heart.gif[/img] J[img]/images/graemlins/heart.gif[/img].

[img]/images/graemlins/diamond.gif[/img] <-> [img]/images/graemlins/club.gif[/img] applied to A[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/spade.gif[/img] vs. Q[img]/images/graemlins/spade.gif[/img] J[img]/images/graemlins/spade.gif[/img] results in A[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/spade.gif[/img] vs. Q[img]/images/graemlins/spade.gif[/img] J[img]/images/graemlins/spade.gif[/img].

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The idea here is to quickly discover all members of each equivalence class for the N-way case. Counting the classes was just a sanity check on my method.

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You can always apply the 24 permutations of the suits to get a complete list of equivalent match-ups. Sometimes there will be only 3 distinct match-ups repeated 8 times. Sometimes there will be 24 distinct collections of hands.

Re: unique showdown probabilities for two preflop hand matchups
 

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However, while your way of organizing the permutations is nice on paper, it looks problematic from an algorithmic perspective.

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I still don't see what difficulties there are and what alternative you offer.

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For example, given the matchup:

A1K1 vs. Q1J1

the "switch two suits" permutation doesn't apply. There aren't two suits to switch.

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There are always 4 suits. You can always apply all 24 permutations of the 4 suits.

[img]/images/graemlins/spade.gif[/img] <-> [img]/images/graemlins/heart.gif[/img] applied to A[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/spade.gif[/img] vs. Q[img]/images/graemlins/spade.gif[/img] J[img]/images/graemlins/spade.gif[/img] results in A[img]/images/graemlins/heart.gif[/img] K[img]/images/graemlins/heart.gif[/img] vs. Q[img]/images/graemlins/heart.gif[/img] J[img]/images/graemlins/heart.gif[/img].

[img]/images/graemlins/diamond.gif[/img] <-> [img]/images/graemlins/club.gif[/img] applied to A[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/spade.gif[/img] vs. Q[img]/images/graemlins/spade.gif[/img] J[img]/images/graemlins/spade.gif[/img] results in A[img]/images/graemlins/spade.gif[/img] K[img]/images/graemlins/spade.gif[/img] vs. Q[img]/images/graemlins/spade.gif[/img] J[img]/images/graemlins/spade.gif[/img].


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Ok, I thought "switch two suits" meant A1K2 vs Q1J2 -> A2K1 vs.Q2J1, which clearly wouldn't make sense if there was only 1 suit in the hand. The language here is ambiguous.

Now I wonder if I know what you meant by "cycle 3 suits." Is it:

A1K2 vs Q3J4 -> A3K1 vs. Q2J4, as in 1,2,3 are "cycled" to 3,2,1?

eastbay

Re: unique showdown probabilities for two preflop hand matchups
 

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Now I wonder if I know what you meant by "cycle 3 suits."

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There are 8 ways to cycle 3 suits. One of them is described by [img]/images/graemlins/spade.gif[/img] -> [img]/images/graemlins/heart.gif[/img] -> [img]/images/graemlins/diamond.gif[/img] -> [img]/images/graemlins/spade.gif[/img]. That means the new matchup has a [img]/images/graemlins/heart.gif[/img] where the original matchup had a [img]/images/graemlins/spade.gif[/img], a [img]/images/graemlins/diamond.gif[/img] where the original had a [img]/images/graemlins/heart.gif[/img], and a [img]/images/graemlins/spade.gif[/img] in place of every [img]/images/graemlins/diamond.gif[/img]. Every [img]/images/graemlins/club.gif[/img] stays the same.

This is how the permutations of {[img]/images/graemlins/spade.gif[/img], [img]/images/graemlins/heart.gif[/img], [img]/images/graemlins/diamond.gif[/img], [img]/images/graemlins/club.gif[/img]} act on a deck of cards or the set of pairs of hands. The 24 permutations of the suits have the structure of a group. The action on the set of pairs of hands has the structure of a group action.