Combinatorial Functions

1
Combinatorial Functions

• Recursion for problem solving

2
Enumerating Initial segments (prefixes)

• inits 1,2,3 ,1,1,2,1,2,3
• inits 2,3 , 2,
  2,3
• fun inits
• inits (xxs)
• (map (fn ys gt xys)
• (inits xs)
• )
• fun inits
• inits xs (inits (init xs))_at_xs

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 (xxs)
• let val ss subseq xs
• in ss_at_(map (fn ys gt xys) ss)
• end

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 (yys) (xyys)
• (map(fn xsgtyxs)(interleave x ys))
• fun perms
• perms (xxs)
• foldr (op _at_)
• (map (interleave x) (perms xs))

5
List partitions

• The list of non-empty lists L1,L2,,Lm forms a
  list-partition of a list L iff
• concat L1,L2,,Lm L
• 1,2 -gt 1,2, 1,2
• 0,1,2 -gt 0,1,2,
• 0,1,2, 0,1,2,
  0,1,2

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 gt 0 )
• Property
• cnt_lp (length xs)
• (length (list_partition xs))

7
Constructing List Partitions

• fun lp
• lp (x) x
• lp (yxxs)
• let val aux lp (xxs) in
• (map (fn ss gt yss) aux)
• _at_
• (map (fn ss gt (y(hd ss))
• (tl ss)) aux)
• end

8
Set partition

• The set of (non-empty) sets s1,s2,,sm
  forms a set partition of s iff the sets sis
  are collectively exhaustive and
  pairwise-disjoint.
• E.g., set partitions of 1,2,3 -gt
• 1,2,3,
• 1, 2,3,
• 1,2, 3,
• 2, 1,3,
• 1,2,3
• (Cf. list partition, number partition, etc.)

9
Divide and Conquer
m-partition of 1,2,3
1 occurring 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
10
Counting problem

• fun cnt_m_sp 1 n 1
• cnt_m_sp m n
• if m gt n then 0
• else if 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

11
(contd)

• upto 3 6 3,4,5,6
• fun upto m n
• if (m gt n) then
• else if (m n) then m
• else m upto (m1) n
• fun cnt_sp n
• foldr (op ) 0
• (map (fn m gt cnt_m_sp m n)
  (upto 1 n))

12
Constructing set partitions

• fun set_parts s
• foldr (op_at_)
• ( map (fn m gt (m_set_parts m s))
• (upto 1 (length s)) )
• fun ins_all e
• ins_all e (sss)
• (((es)ss)
• ( map (fn ts gt s ts)
• (ins_all e ss) ) )

13

• fun m_set_parts 1 s s
• m_set_parts m (s as hsts)
• let val n (length s) in
• if m gt n then
• else if m n then
• foldr (fn (e,ss)gtess ) s
• else let
• val p1 (m_set_parts (m-1) ts)
• val p2 (m_set_parts m ts)
• in
• (map (fn ss gt hsss) p1)
• _at_
• (foldr (op_at_) (map (ins_all hs) p2 ))
• end
• end