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Combinatorial Functions

Explore the comprehensive world of combinatorial functions focusing on the enumeration of initial segments (prefixes), subsequences, permutations, and list/set partitions. Discover methods for generating these structures using recursion and functional programming principles. The guide illustrates the processes behind each combinatorial concept, including counting partitions, set partitions, and automate techniques for dividing lists and sets into all possible configurations. This resource is essential for problem solvers and programmers interested in adopting combinatorial strategies in their algorithms.

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Combinatorial Functions

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  1. Combinatorial Functions Recursion for problem solving L10-COMB-FN

  2. Enumerating Initial segments (prefixes) inits [1,2,3] = [[],[1],[1,2],[1,2,3]] inits [2,3] = [ [], [2], [2,3] ] fun inits [] = [[]] | inits (x::xs) = []:: (map (fn ys => x::ys) (inits xs) ); fun inits [] = [[]] | inits xs = (inits (init xs))@[xs]; L10-COMB-FN

  3. Enumerating Subsequences subseq [2,3] = [[],[2],[3],[2,3]]; subseq [1,2,3] = [[], [1],[2],[3], [1,2],[1,3],[2,3],[1,2,3] ]; fun subseq [] = [[]] | subseq (x::xs) = letval ss = subseq xs in ss@(map (fn ys => x::ys) ss) end; L10-COMB-FN

  4. Enumerating permutations perms [2] = [[2]] perms [1,2] = [[1,2], [2,1]] perms [0,1,2] = [[0,1,2],[1,0,2],[1,2,0], [0,2,1],[2,0,1],[2,1,0]] fun interleave x [] = [[x]] | interleave x (y::ys) = (x::y::ys) :: (map(fn xs=>y::xs)(interleave x ys)); fun perms [] = [[]] | perms (x::xs) = foldr (op @) [] (map (interleave x) (perms xs)) ; L10-COMB-FN

  5. List partitions • The list of non-empty lists [L1,L2,…,Lm] forms a list-partition of a list Liff concat [L1,L2,…,Lm] = L [1,2] -> [[[1,2]], [[1],[2]]] [0,1,2] -> [ [[0,1,2]], [[0,1],[2]], [[0],[1,2]], [[0],[1],[2]] ] L10-COMB-FN

  6. Counting problem fun cnt_lp 0 = 0 | cnt_lp 1 = 1 | cnt_lp n = 2 * (cnt_lp (n-1)); (* cnt_lp n = 2^(n-1) for n > 0 *) Property: cnt_lp (length xs) = (length (list_partition xs)) L10-COMB-FN

  7. Constructing List Partitions fun lp [] = [] | lp (x::[]) = [ [[x]] ] | lp (y::x::xs) = letval aux = lp (x::xs) in (map (fn ss => [y]::ss) aux) @ (map (fn ss => (y::(hd ss)) :: (tl ss)) aux) end; L10-COMB-FN

  8. Set partition • The set of (non-empty) sets [s1,s2,…,sm] forms a set partition of siff the sets si’s are collectively exhaustive and pairwise-disjoint. • E.g., set partitions of{1,2,3} -> { {{1,2,3}}, { {1}, {2,3}}, {{1,2}, {3}}, {{2}, {1,3}}, { {1},{2},{3}} } (Cf. list partition, number partition, etc.) L10-COMB-FN

  9. Divide and Conquer m-partition of {1,2,3} 1occurring with with a part in m-partition of {2,3} (m-1)-partition of {2,3} solitary part{1}:: 2-partitions of {1,2,3} = {{ {1},{2,3}} } U {{{1,2},{3} } , { {2},{1,3} }} L10-COMB-FN

  10. Counting problem fun cnt_m_sp 1 n = 1 | cnt_m_sp m n = if m > n then 0 elseif m = n then 1 else (cnt_m_sp (m-1) (n-1)) + (m * (cnt_m_sp m (n-1))); • Dependency (visualization) • Basis: Row 1 and diagonal (Half-plane inapplicable) • Recursive step: Previous row, previous column L10-COMB-FN

  11. (cont’d) upto 3 6 = [3,4,5,6] fun upto m n = if (m > n) then [] else if (m = n) then [m] else m:: upto (m+1) n; fun cnt_sp n = foldr (op +) 0 (map (fn m => cnt_m_sp m n) (upto 1 n)); L10-COMB-FN

  12. Constructing set partitions fun set_parts s = foldr (op@) [] ( map (fn m => (m_set_parts m s)) (upto 1 (length s)) ); fun ins_all e [] = [] | ins_all e (s::ss) = (((e::s)::ss) :: ( map (fnts => s :: ts) (ins_all e ss))); L10-COMB-FN

  13. fun m_set_parts 1 s = [[s]] | m_set_parts m (s as hs::ts)= letval n = (length s) in if m > n then [] else if m = n then [foldr (fn (e,ss)=>[e]::ss ) [] s] elselet val p1 = (m_set_parts (m-1) ts) val p2 = (m_set_parts m ts) in (map (fn ss => [hs]::ss) p1) @ (foldr (op@) [](map (ins_all hs) p2 )) end end; L10-COMB-FN

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