function count = poly_coef_count ( dim, degree )
%*****************************************************************************80
%
%% POLY_COEF_COUNT: polynomial coefficient count given dimension and degree.
%
% Discussion:
%
% To count all monomials of degree 5 or less in dimension 3,
% we can count all monomials of degree 5 in dimension 4.
%
% To count all monomials of degree 5 in dimension 4, we imagine
% that each of the variables X, Y, Z and W is a "box" and that
% we need to drop 5 pebbles into these boxes. Every distinct
% way of doing this represents a degree 5 monomial in dimension 4.
% Ignoring W gives us monomials up to degree five in dimension 3.
%
% To count them, we draw 3 lines as separators to indicate the
% 4 boxes, and then imagine all disctinct sequences involving
% the three lines and the 5 pebbles. Indicate the lines by 1's
% and the pebbles by 0's and we're asking for the number of
% permutations of 3 1's and 5 0's, which is 8! / (3! 5!)
%
% In other words, 56 = 8! / (3! 5!) is:
% * the number of monomials of degree exactly 5 in dimension 4,
% * the number of monomials of degree 5 or less in dimension 3,
% * the number of polynomial coefficients of a polynomial of
% degree 5 in (X,Y,Z).
%
% In general, the formula for the number of monomials of degree DEG
% or less in dimension DIM is
%
% (DEG+DIM)! / (DEG! * DIM!)
%
% Licensing:
%
% This code is distributed under the MIT license.
%
% Modified:
%
% 11 January 2021
%
% Author:
%
% John Burkardt
%
% Input:
%
% integer DIM, the dimension of the polynomial.
% 0 <= DIM.
%
% integer DEGREE, the degree of the polynomnial
% 0 <= DEGREE
%
% Output:
%
% integer COUNT, the number of coefficients
% in the general polynomial of dimension DIM and degree DEGREE.
%
if ( dim < 0 )
count = -1;
elseif ( degree < 0 )
count = -1;
else
count = nchoosek ( degree + dim, degree );
end
return
end