Short F# script to generate and count the number of simultaneously symmetric and anti-symmetric relations on a set A = {1, 2, 3, 4, 5}

relations.fsx

module List =
    // Some code for generating permutations of a list
    // Please disregard it, I copied it off the internet :)
    let rec permutations l =
        let rec aux = function
            | []      -> seq [List.empty]
            | x :: xs -> Seq.collect (insertions x) (aux xs)
        aux l |> Seq.toList
    and insertions x = function
        | []             -> [[x]]
        | (y :: ys) as xs -> (x::xs)::(List.map (fun x -> y::x) (insertions x ys))

// Define a function on an integer n
// that generates a list of lists, each
// containing the elements of a relation
// of size n which is both symmetric
// and anti-symmetric
let relationsOfSize n =
    [1..5]                      // For the list of numbers from 1 through 5 (similar to our set A)
    |> List.permutations        // Generate a list of permutations
    |> List.map (List.take n    // Take the first n elements of each permutation
    >> List.sort)               // ... and sort them all
    |> List.distinct            // Remove all the duplicates

// For the sake of outputting some interesting
// information, print out the various relations
// as well as the number of relations of each size
// from 0 through 5
[0..5]
|> List.map (fun i ->
    let r = relationsOfSize i
    printfn "Relations of size %i: %A\nQuantity: %i" i r (List.length r))

// Compute the max number of simultaneously
// symmetric and anti-symmetric relations
// of sizes 0 through 5 on a set of numbers 1
// through 5.
let maxRelations =
    [0..5]                      // For the list of numbers 0 through 5
    |> List.map relationsOfSize // Generate all simultaneously symmetric and anti-symmetric relations of corresponding size
    |> List.concat              // Combine the lists of lists of relations into one list of relations
    |> List.length              // and output the resulting list's length

// Print out the max number of relations, should be 32
printfn "%i" maxRelations