Quiz (Test MathJax)
Published:
We consider polynomials whose degree and coefficients satisfy a natural number constraint involving their sum. By fixing the degree, the problem reduces to counting compositions of an integer into positive parts, which can be solved using the stars-and-bars method. The number of solutions for each fixed degree corresponds to a binomial coefficient, and summing over all possible degrees yields a power of two. Consequently, the total number of admissible polynomials is given by an exponential expression.
Quiz
We investigate the enumeration of polynomials of the form
\[M(x) = \sum_{k=0}^{n} a_k x^k\]where all coefficients except the leading one are nonnegative integers, and the leading coefficient is a positive integer, under the constraint
\[n + \sum_{k=0}^{n} a_k = t.\]For each fixed degree, the problem reduces to counting the number of compositions of an integer into positive parts, which can be computed using the stars-and-bars method. The number of solutions for each fixed degree is given by a binomial coefficient. Summing over all possible degrees yields
\[\sum_{n=0}^{t-1} \binom{t-1}{n} = 2^{t-1}.\]Therefore, the total number of admissible polynomials is
\[2^{t-1}.\]