Many sequences *p _{n}(x)* of polynomials seem to
have interesting sets of (complex) zeros. Here are some examples.

- Let
*K*denote the sum over all connected simple graphs on an_{n}(x)*n*-element vertex set of*x*, where^{e(G)-n+1}*e(G)*denotes the number of edges of*G*. For instance,*K*= 3 +_{3}(x)*x*. Zeros of*K*: ps or pdf._{14}(x)

- Let
*X*denote the chromatic polynomial of the complete bipartite graph_{m,n}(q)*K*. Equivalently,_{m,n}*X*is the coefficient of_{m,n}(q)*x*in the power series expansion about^{m}y^{n}/m!n!*x*=*y*=0 of*(e*. Zeros of^{x}+ e^{y}-1)^{q}*X*: ps or pdf. Note. Appearances to the contrary, the integers 2, 3, 4, 5, 6, 7, 8, 9 are not zeros of_{20,20}(q)*X*. For instance, the zero close to_{20,20}(q)*q*=2 is actually 2.0000000000000000000000000000000000000391710202489...

- Let
*C*denote the sum of_{n}(q)*q*over all lattice paths^{A(P)}*P*from (0,0) to (*n,n*) with steps (0,1) and (1,0), such that*P*never rises above the line*y=x*, and where*A(P)*is the area under the path (and above the*x*-axis). The number of such paths is the Catalan number*C*, and_{n}*C*is a standard_{n}(q)*q*-analogue of*C*(see_{n}*Enumerative Combinatorics*, vol. 2, Exercise 6.34(a)). For instance,*C*. Zeros of_{3}(q) = 1 + q + 2q^{2}+ q^{3}*C*: ps or pdf._{32}(q)

- Let
*H*denote the number of_{n}(r)*n*×*n*matrices of nonnegative integers such that every row and column sums to*r*. For instance,*H*=1,_{n}(0)*H*, and_{n}(1)=n!*H*. It is known that for fixed_{2}(r)=r+1*n*,*H*is a polynomial in_{n}(r)*r*of degree*(n-1)*. (See for instance^{2}*Enumerative Combinatorics*, vol. 1, Proposition 4.6.19.) Zeros of*H*: ps or pdf._{9}(r)

- (suggested by G. Labelle) Let
*N*be the sum of (1/n)φ_{n}(x)*(d)x*over all (positive) divisors^{n/d}*d*of*n*, where φ*(d)*denote Euler's phi-function. If*x*is a positive integer, then*N*is the number of "necklaces" of length_{n}(x)*n*with*x*colors, i.e., the number of sequences, up to cyclic shift, with*n*terms from an alphabet of*x*letters. Zeros of*N*: ps or pdf. Similar behavior is seen when φ is replaced by μ (the Möbius function of number theory), in which case necklaces are replaced by_{120}(x)*primitive*necklaces (those with no symmetry).

- (suggested by H. Wilf) Let
*B*denote the_{n}(x)*n*th Bernoulli polynomial. Zeros of*B*: pdf (_{300}(x)**corrected**on 5/2/04). Click here (pdf file) for information on the*real*zeros of*B*. For the limiting curve formed by the complex zeros, click here._{n}(x)

- Let
*p*denote the number of partitions of the positive integer_{k}(n)*n*into*k*parts. Let*F*denote the sum over all_{k}(x)*n*of*p*. For instance,_{k}(n)x^{n}*F*=_{4}(x)*x*+^{4}*x*+2^{3}*x*+^{2}*x*. Zeros of*F*: ps or pdf._{200}(x)

- Zeros of the sum from 1 to 300 of
*x*/(sin^{n}*n*).