// output of ./demo/comb/composition-nz-minc-demo.cc:
// Description:
//% Compositions of n into positive parts with first part c and
//%   each part <= f times its predecessor.
//% For c=1 the same as: f-ary Huffman codes (canonical trees) with
//%   (f-1)*n+1 terminal nodes, a[k] is the number of internal nodes of depth k.
//% Such compositions (for f=2) are treated in
//%   Henryk Minc: "A Problem in Partitions: Enumeration of Elements of a
//%     given Degree in the free commutative entropic cyclic Groupoid",
//%   Proceedings of the Edinburgh Mathematical Society (2),
//%   vol.11, no.4, pp.223-224, (November-1959).
//% The compositions for f=2 are also called "Cayley compositions", see
//%   George E. Andrews, Peter Paule, Axel Riese, Volker Strehl:
//%   "MacMahon's Partition Analysis V: Bijections, Recursions, and Magic Squares",
//%   in: Algebraic Combinatorics and Applications, proceedings of Euroconference Alcoma 99,
//%   September 12-19, 1999, Goessweinstein, Germany,
//%   A. Betten, A. Kohnert, R. Laue, A. Wassermann eds., Springer-Verlag, pp.1-39, (2001).
//% See also:
//%   Christian Elsholtz, Clemens Heuberger, Helmut Prodinger:
//%   "The number of Huffman codes, compact trees, and sums of unit fractions",
//%   arXiv:1108.5964 [math.CO], (30-August-2011).
//%
//% Cf. OEIS sequences:
//% f=2:  A002572 (c=1), A002573 (c=2), A002574 (c=3),
//%       A049284 (c=4), A049285 (c=5).
//% c=1:  A002572 (f=2), A176485 (f=3), A176503 (f=4),
//%       A194628 (f=5), A194629 (f=6), A194630 (f=7),
//%       A194631 (f=8), A194632 (f=9), A194633 (f=10).

arg 1: 9 == n  [compositions of n]  default=9
arg 2: 1 == c  [first part]  default=1
arg 3: 2 == f  [max. ratio of consecutive parts]  default=2
   0:    [ . 1 1 1 1 1 1 1 1 2 ]  [ 1 1 1 1 1 1 1 1 1 ]
   1:    [ . 1 1 1 1 1 1 . 4 . ]  [ 1 1 1 1 1 1 1 2 ]
   2:    [ . 1 1 1 1 1 . 3 2 . ]  [ 1 1 1 1 1 1 2 1 ]
   3:    [ . 1 1 1 1 . 3 1 2 . ]  [ 1 1 1 1 1 2 1 1 ]
   4:    [ . 1 1 1 1 . 2 4 . . ]  [ 1 1 1 1 1 2 2 ]
   5:    [ . 1 1 1 . 3 1 1 2 . ]  [ 1 1 1 1 2 1 1 1 ]
   6:    [ . 1 1 1 . 3 . 4 . . ]  [ 1 1 1 1 2 1 2 ]
   7:    [ . 1 1 1 . 2 3 2 . . ]  [ 1 1 1 1 2 2 1 ]
   8:    [ . 1 1 1 . 1 6 . . . ]  [ 1 1 1 1 2 3 ]
   9:    [ . 1 1 . 3 1 1 1 2 . ]  [ 1 1 1 2 1 1 1 1 ]
  10:    [ . 1 1 . 3 1 . 4 . . ]  [ 1 1 1 2 1 1 2 ]
  11:    [ . 1 1 . 3 . 3 2 . . ]  [ 1 1 1 2 1 2 1 ]
  12:    [ . 1 1 . 2 3 1 2 . . ]  [ 1 1 1 2 2 1 1 ]
  13:    [ . 1 1 . 2 2 4 . . . ]  [ 1 1 1 2 2 2 ]
  14:    [ . 1 1 . 1 5 2 . . . ]  [ 1 1 1 2 3 1 ]
  15:    [ . 1 1 . . 8 . . . . ]  [ 1 1 1 2 4 ]
  16:    [ . 1 . 3 1 1 1 1 2 . ]  [ 1 1 2 1 1 1 1 1 ]
  17:    [ . 1 . 3 1 1 . 4 . . ]  [ 1 1 2 1 1 1 2 ]
  18:    [ . 1 . 3 1 . 3 2 . . ]  [ 1 1 2 1 1 2 1 ]
  19:    [ . 1 . 3 . 3 1 2 . . ]  [ 1 1 2 1 2 1 1 ]
  20:    [ . 1 . 3 . 2 4 . . . ]  [ 1 1 2 1 2 2 ]
  21:    [ . 1 . 2 3 1 1 2 . . ]  [ 1 1 2 2 1 1 1 ]
  22:    [ . 1 . 2 3 . 4 . . . ]  [ 1 1 2 2 1 2 ]
  23:    [ . 1 . 2 2 3 2 . . . ]  [ 1 1 2 2 2 1 ]
  24:    [ . 1 . 2 1 6 . . . . ]  [ 1 1 2 2 3 ]
  25:    [ . 1 . 1 5 1 2 . . . ]  [ 1 1 2 3 1 1 ]
  26:    [ . 1 . 1 4 4 . . . . ]  [ 1 1 2 3 2 ]
  27:    [ . 1 . . 7 2 . . . . ]  [ 1 1 2 4 1 ]
  28:    [ . . 3 1 1 1 1 1 2 . ]  [ 1 2 1 1 1 1 1 1 ]
  29:    [ . . 3 1 1 1 . 4 . . ]  [ 1 2 1 1 1 1 2 ]
  30:    [ . . 3 1 1 . 3 2 . . ]  [ 1 2 1 1 1 2 1 ]
  31:    [ . . 3 1 . 3 1 2 . . ]  [ 1 2 1 1 2 1 1 ]
  32:    [ . . 3 1 . 2 4 . . . ]  [ 1 2 1 1 2 2 ]
  33:    [ . . 3 . 3 1 1 2 . . ]  [ 1 2 1 2 1 1 1 ]
  34:    [ . . 3 . 3 . 4 . . . ]  [ 1 2 1 2 1 2 ]
  35:    [ . . 3 . 2 3 2 . . . ]  [ 1 2 1 2 2 1 ]
  36:    [ . . 3 . 1 6 . . . . ]  [ 1 2 1 2 3 ]
  37:    [ . . 2 3 1 1 1 2 . . ]  [ 1 2 2 1 1 1 1 ]
  38:    [ . . 2 3 1 . 4 . . . ]  [ 1 2 2 1 1 2 ]
  39:    [ . . 2 3 . 3 2 . . . ]  [ 1 2 2 1 2 1 ]
  40:    [ . . 2 2 3 1 2 . . . ]  [ 1 2 2 2 1 1 ]
  41:    [ . . 2 2 2 4 . . . . ]  [ 1 2 2 2 2 ]
  42:    [ . . 2 1 5 2 . . . . ]  [ 1 2 2 3 1 ]
  43:    [ . . 2 . 8 . . . . . ]  [ 1 2 2 4 ]
  44:    [ . . 1 5 1 1 2 . . . ]  [ 1 2 3 1 1 1 ]
  45:    [ . . 1 5 . 4 . . . . ]  [ 1 2 3 1 2 ]
  46:    [ . . 1 4 3 2 . . . . ]  [ 1 2 3 2 1 ]
  47:    [ . . 1 3 6 . . . . . ]  [ 1 2 3 3 ]
  48:    [ . . . 7 1 2 . . . . ]  [ 1 2 4 1 1 ]
  49:    [ . . . 6 4 . . . . . ]  [ 1 2 4 2 ]
 ct=50