function [ n_data, n, c ] = partition_distinct_count_values ( n_data )
%*****************************************************************************80
%
%% partition_distinct_count_values() returns some values of Q(N).
%
% Discussion:
%
% A partition of an integer N is a representation of the integer
% as the sum of nonzero positive integers. The order of the summands
% does not matter. The number of partitions of N is symbolized
% by P(N). Thus, the number 5 has P(N) = 7, because it has the
% following partitions:
%
% 5 = 5
% = 4 + 1
% = 3 + 2
% = 3 + 1 + 1
% = 2 + 2 + 1
% = 2 + 1 + 1 + 1
% = 1 + 1 + 1 + 1 + 1
%
% However, if we require that each member of the partition
% be distinct, so that no nonzero summand occurs more than once,
% we are computing something symbolized by Q(N).
% The number 5 has Q(N) = 3, because it has the following partitions
% into distinct parts:
%
% 5 = 5
% = 4 + 1
% = 3 + 2
%
% In Mathematica, the function can be evaluated by
%
% PartitionsQ[n]
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 19 September 2004
%
% Author:
%
% John Burkardt
%
% Reference:
%
% Milton Abramowitz and Irene Stegun,
% Handbook of Mathematical Functions,
% US Department of Commerce, 1964.
%
% Stephen Wolfram,
% The Mathematica Book,
% Fourth Edition,
% Wolfram Media / Cambridge University Press, 1999.
%
% Input:
%
% integer N_DATA. The user sets N_DATA to 0 before the first call.
% Thereafter, it should simply be the value returned by the previous call.
%
% Output:
%
% integer N_DATA. On each call, the routine increments N_DATA by 1, and
% returns the corresponding data; when there is no more data, the
% output value of N_DATA will be 0 again.
%
% integer N, the integer.
%
% integer C, the number of partitions of the integer
% into distinct parts.
%
n_max = 21;
c_vec = [ ...
1, ...
1, 1, 2, 2, 3, 4, 5, 6, 8, 10, ...
12, 15, 18, 22, 27, 32, 38, 46, 54, 64 ];
n_vec = [ ...
0, ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
11, 12, 13, 14, 15, 16, 17, 18, 19, 20 ];
if ( n_data < 0 )
n_data = 0;
end
n_data = n_data + 1;
if ( n_max < n_data )
n_data = 0;
n = 0;
c = 0;
else
n = n_vec(n_data);
c = c_vec(n_data);
end
return
end