// output of ./demo/comb/perm-genus-demo.cc:
// Description:
//% Genus of all permutations of n elements.
//% Print parenthesis strings for permutations of genus zero.

arg 1: 4 == n  [Permutations of n elements.]  default=4
   0:    [ . 1 2 3 ]  0   (0) (1) (2) (3)    (0, 1, 2, 3)    1.1.1.1.   ()()()()
   1:    [ . 1 3 2 ]  0   (0) (1) (2, 3)     (0, 1, 3) (2)   1.1.11..   ()()(())
   2:    [ . 2 1 3 ]  0   (0) (1, 2) (3)     (0, 2, 3) (1)   1.11..1.   ()(())()
   3:    [ . 2 3 1 ]  0   (0) (1, 2, 3)      (0, 3) (1) (2)  1.11.1..   ()(()())
   4:    [ . 3 1 2 ]  1   (0) (1, 3, 2)      (0, 2, 1, 3)    
   5:    [ . 3 2 1 ]  0   (0) (1, 3) (2)     (0, 3) (1, 2)   1.111...   ()((()))
   6:    [ 1 . 2 3 ]  0   (0, 1) (2) (3)     (0) (1, 2, 3)   11..1.1.   (())()()
   7:    [ 1 . 3 2 ]  0   (0, 1) (2, 3)      (0) (1, 3) (2)  11..11..   (())(())
   8:    [ 1 2 . 3 ]  0   (0, 1, 2) (3)      (0) (1) (2, 3)  11.1..1.   (()())()
   9:    [ 1 2 3 . ]  0   (0, 1, 2, 3)       (0) (1) (2) (3) 11.1.1..   (()()())
  10:    [ 1 3 . 2 ]  1   (0, 1, 3, 2)       (0) (1, 3, 2)   
  11:    [ 1 3 2 . ]  0   (0, 1, 3) (2)      (0) (1, 2) (3)  11.11...   (()(()))
  12:    [ 2 . 1 3 ]  1   (0, 2, 1) (3)      (0, 2, 3, 1)    
  13:    [ 2 . 3 1 ]  1   (0, 2, 3, 1)       (0, 3, 1) (2)   
  14:    [ 2 1 . 3 ]  0   (0, 2) (1) (3)     (0, 1) (2, 3)   111...1.   ((()))()
  15:    [ 2 1 3 . ]  0   (0, 2, 3) (1)      (0, 1) (2) (3)  111..1..   ((())())
  16:    [ 2 3 . 1 ]  1   (0, 2) (1, 3)      (0, 3, 2, 1)    
  17:    [ 2 3 1 . ]  1   (0, 2, 1, 3)       (0, 2, 1) (3)   
  18:    [ 3 . 1 2 ]  1   (0, 3, 2, 1)       (0, 2) (1, 3)   
  19:    [ 3 . 2 1 ]  1   (0, 3, 1) (2)      (0, 3, 1, 2)    
  20:    [ 3 1 . 2 ]  1   (0, 3, 2) (1)      (0, 1, 3, 2)    
  21:    [ 3 1 2 . ]  0   (0, 3) (1) (2)     (0, 1, 2) (3)   111.1...   ((()()))
  22:    [ 3 2 . 1 ]  1   (0, 3, 1, 2)       (0, 3, 2) (1)   
  23:    [ 3 2 1 . ]  0   (0, 3) (1, 2)      (0, 2) (1) (3)  1111....   (((())))
  ct=24
14, 10,