// output of ./demo/comb/involution-zero-map-rgs-demo.cc:
// Description:
//% Restricted growth strings (RGS):
//%   each digit a[k] is either zero or a[k] < k, a[a[k]] == 0,
//%   and there is at most one digit a[k] in the RGS.
//% Same as: maps from {1, 2, 3, ..., n} to {0, 1, 2, 3, ..., n}
//%   such that f(x) < x and f(f(x)) == 0 and there is no t!=x with f(t) = f(x).
//% Lexicographic order.
//% Cf. OEIS sequence A000085.

arg 1: 6 == n  [RGS of length n]  default=6
  1:    [ . . . . . . ]    [ . . . . . . ]    (0), (1), (2), (3), (4), (5)
  2:    [ . . . . . 1 ]    [ . . . . . 5 ]    (0, 5), (1), (2), (3), (4)
  3:    [ . . . . . 2 ]    [ . . . . . 4 ]    (0), (1, 5), (2), (3), (4)
  4:    [ . . . . . 3 ]    [ . . . . . 3 ]    (0), (1), (2, 5), (3), (4)
  5:    [ . . . . . 4 ]    [ . . . . . 2 ]    (0), (1), (2), (3, 5), (4)
  6:    [ . . . . . 5 ]    [ . . . . . 1 ]    (0), (1), (2), (3), (4, 5)
  7:    [ . . . . 1 . ]    [ . . . . 4 . ]    (0, 4), (1), (2), (3), (5)
  8:    [ . . . . 1 2 ]    [ . . . . 4 4 ]    (0, 4), (1, 5), (2), (3)
  9:    [ . . . . 1 3 ]    [ . . . . 4 3 ]    (0, 4), (1), (2, 5), (3)
 10:    [ . . . . 1 4 ]    [ . . . . 4 2 ]    (0, 4), (1), (2), (3, 5)
 11:    [ . . . . 2 . ]    [ . . . . 3 . ]    (0), (1, 4), (2), (3), (5)
 12:    [ . . . . 2 1 ]    [ . . . . 3 5 ]    (0, 5), (1, 4), (2), (3)
 13:    [ . . . . 2 3 ]    [ . . . . 3 3 ]    (0), (1, 4), (2, 5), (3)
 14:    [ . . . . 2 4 ]    [ . . . . 3 2 ]    (0), (1, 4), (2), (3, 5)
 15:    [ . . . . 3 . ]    [ . . . . 2 . ]    (0), (1), (2, 4), (3), (5)
 16:    [ . . . . 3 1 ]    [ . . . . 2 5 ]    (0, 5), (1), (2, 4), (3)
 17:    [ . . . . 3 2 ]    [ . . . . 2 4 ]    (0), (1, 5), (2, 4), (3)
 18:    [ . . . . 3 4 ]    [ . . . . 2 2 ]    (0), (1), (2, 4), (3, 5)
 19:    [ . . . . 4 . ]    [ . . . . 1 . ]    (0), (1), (2), (3, 4), (5)
 20:    [ . . . . 4 1 ]    [ . . . . 1 5 ]    (0, 5), (1), (2), (3, 4)
 21:    [ . . . . 4 2 ]    [ . . . . 1 4 ]    (0), (1, 5), (2), (3, 4)
 22:    [ . . . . 4 3 ]    [ . . . . 1 3 ]    (0), (1), (2, 5), (3, 4)
 23:    [ . . . 1 . . ]    [ . . . 3 . . ]    (0, 3), (1), (2), (4), (5)
 24:    [ . . . 1 . 2 ]    [ . . . 3 . 4 ]    (0, 3), (1, 5), (2), (4)
 25:    [ . . . 1 . 3 ]    [ . . . 3 . 3 ]    (0, 3), (1), (2, 5), (4)
 26:    [ . . . 1 . 5 ]    [ . . . 3 . 1 ]    (0, 3), (1), (2), (4, 5)
 27:    [ . . . 1 2 . ]    [ . . . 3 3 . ]    (0, 3), (1, 4), (2), (5)
 28:    [ . . . 1 2 3 ]    [ . . . 3 3 3 ]    (0, 3), (1, 4), (2, 5)
 29:    [ . . . 1 3 . ]    [ . . . 3 2 . ]    (0, 3), (1), (2, 4), (5)
 30:    [ . . . 1 3 2 ]    [ . . . 3 2 4 ]    (0, 3), (1, 5), (2, 4)
 31:    [ . . . 2 . . ]    [ . . . 2 . . ]    (0), (1, 3), (2), (4), (5)
 32:    [ . . . 2 . 1 ]    [ . . . 2 . 5 ]    (0, 5), (1, 3), (2), (4)
 33:    [ . . . 2 . 3 ]    [ . . . 2 . 3 ]    (0), (1, 3), (2, 5), (4)
 34:    [ . . . 2 . 5 ]    [ . . . 2 . 1 ]    (0), (1, 3), (2), (4, 5)
 35:    [ . . . 2 1 . ]    [ . . . 2 4 . ]    (0, 4), (1, 3), (2), (5)
 36:    [ . . . 2 1 3 ]    [ . . . 2 4 3 ]    (0, 4), (1, 3), (2, 5)
 37:    [ . . . 2 3 . ]    [ . . . 2 2 . ]    (0), (1, 3), (2, 4), (5)
 38:    [ . . . 2 3 1 ]    [ . . . 2 2 5 ]    (0, 5), (1, 3), (2, 4)
 39:    [ . . . 3 . . ]    [ . . . 1 . . ]    (0), (1), (2, 3), (4), (5)
 40:    [ . . . 3 . 1 ]    [ . . . 1 . 5 ]    (0, 5), (1), (2, 3), (4)
 41:    [ . . . 3 . 2 ]    [ . . . 1 . 4 ]    (0), (1, 5), (2, 3), (4)
 42:    [ . . . 3 . 5 ]    [ . . . 1 . 1 ]    (0), (1), (2, 3), (4, 5)
 43:    [ . . . 3 1 . ]    [ . . . 1 4 . ]    (0, 4), (1), (2, 3), (5)
 44:    [ . . . 3 1 2 ]    [ . . . 1 4 4 ]    (0, 4), (1, 5), (2, 3)
 45:    [ . . . 3 2 . ]    [ . . . 1 3 . ]    (0), (1, 4), (2, 3), (5)
 46:    [ . . . 3 2 1 ]    [ . . . 1 3 5 ]    (0, 5), (1, 4), (2, 3)
 47:    [ . . 1 . . . ]    [ . . 2 . . . ]    (0, 2), (1), (3), (4), (5)
 48:    [ . . 1 . . 2 ]    [ . . 2 . . 4 ]    (0, 2), (1, 5), (3), (4)
 49:    [ . . 1 . . 4 ]    [ . . 2 . . 2 ]    (0, 2), (1), (3, 5), (4)
 50:    [ . . 1 . . 5 ]    [ . . 2 . . 1 ]    (0, 2), (1), (3), (4, 5)
 51:    [ . . 1 . 2 . ]    [ . . 2 . 3 . ]    (0, 2), (1, 4), (3), (5)
 52:    [ . . 1 . 2 4 ]    [ . . 2 . 3 2 ]    (0, 2), (1, 4), (3, 5)
 53:    [ . . 1 . 4 . ]    [ . . 2 . 1 . ]    (0, 2), (1), (3, 4), (5)
 54:    [ . . 1 . 4 2 ]    [ . . 2 . 1 4 ]    (0, 2), (1, 5), (3, 4)
 55:    [ . . 1 2 . . ]    [ . . 2 2 . . ]    (0, 2), (1, 3), (4), (5)
 56:    [ . . 1 2 . 5 ]    [ . . 2 2 . 1 ]    (0, 2), (1, 3), (4, 5)
 57:    [ . . 2 . . . ]    [ . . 1 . . . ]    (0), (1, 2), (3), (4), (5)
 58:    [ . . 2 . . 1 ]    [ . . 1 . . 5 ]    (0, 5), (1, 2), (3), (4)
 59:    [ . . 2 . . 4 ]    [ . . 1 . . 2 ]    (0), (1, 2), (3, 5), (4)
 60:    [ . . 2 . . 5 ]    [ . . 1 . . 1 ]    (0), (1, 2), (3), (4, 5)
 61:    [ . . 2 . 1 . ]    [ . . 1 . 4 . ]    (0, 4), (1, 2), (3), (5)
 62:    [ . . 2 . 1 4 ]    [ . . 1 . 4 2 ]    (0, 4), (1, 2), (3, 5)
 63:    [ . . 2 . 4 . ]    [ . . 1 . 1 . ]    (0), (1, 2), (3, 4), (5)
 64:    [ . . 2 . 4 1 ]    [ . . 1 . 1 5 ]    (0, 5), (1, 2), (3, 4)
 65:    [ . . 2 1 . . ]    [ . . 1 3 . . ]    (0, 3), (1, 2), (4), (5)
 66:    [ . . 2 1 . 5 ]    [ . . 1 3 . 1 ]    (0, 3), (1, 2), (4, 5)
 67:    [ . 1 . . . . ]    [ . 1 . . . . ]    (0, 1), (2), (3), (4), (5)
 68:    [ . 1 . . . 3 ]    [ . 1 . . . 3 ]    (0, 1), (2, 5), (3), (4)
 69:    [ . 1 . . . 4 ]    [ . 1 . . . 2 ]    (0, 1), (2), (3, 5), (4)
 70:    [ . 1 . . . 5 ]    [ . 1 . . . 1 ]    (0, 1), (2), (3), (4, 5)
 71:    [ . 1 . . 3 . ]    [ . 1 . . 2 . ]    (0, 1), (2, 4), (3), (5)
 72:    [ . 1 . . 3 4 ]    [ . 1 . . 2 2 ]    (0, 1), (2, 4), (3, 5)
 73:    [ . 1 . . 4 . ]    [ . 1 . . 1 . ]    (0, 1), (2), (3, 4), (5)
 74:    [ . 1 . . 4 3 ]    [ . 1 . . 1 3 ]    (0, 1), (2, 5), (3, 4)
 75:    [ . 1 . 3 . . ]    [ . 1 . 1 . . ]    (0, 1), (2, 3), (4), (5)
 76:    [ . 1 . 3 . 5 ]    [ . 1 . 1 . 1 ]    (0, 1), (2, 3), (4, 5)
 ct=76