Enumerative combinatorics
Inductive and injective proofs of log concavity results
Discrete Mathematics
The partition polynomial of a finite set system
Journal of Combinatorial Theory Series A
Combinatorics and total positivity
Journal of Combinatorial Theory Series A
Regular Article: On Some Numbers Related to Whitney Numbers of Dowling Lattices
Advances in Applied Mathematics
Regular Article: Log-Concavity of Whitney Numbers of Dowling Lattices
Advances in Applied Mathematics
On the unimodality of independence polynomials of some graphs
European Journal of Combinatorics
The real-rootedness and log-concavities of coordinator polynomials of Weyl group lattices
European Journal of Combinatorics
A refined sign-balance of simsun permutations
European Journal of Combinatorics
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Let f(x) and g(x) be two real polynomials whose leading coefficients have the same sign. Suppose that f(x) and g(x) have only real zeros and that g interfaces f or g alternates left of f. We show that if ad ≥ bc then the polynomial (bx + a)f(x) + (dx + c)g(x) has only real zeros. Applications are related to certain results of Brenti (Mem. Amer. Math. Soc. 413 (1989)) and transformations of Pólya-frequency (PF) sequences. More specifically, suppose that A(n, k) are nonnegative numbers which satisfy the recurrence A(n,k) = (rn + sk + t)A(n - 1,k - 1) + (an + bk + c)A(n - 1,k) for n ≥ 1 and 0 ≤ k ≤ n, where A(n,k) = 0 unless 0≤k≤n. We show that if rb≥as and (r+s+t)b≥(a+c)s, then for each n≥0, A(n, 0), A(n, 1),..., A(n, n) is a PF sequence. This gives a unified proof of the PF property of many well-known sequences including the binomial coefficients, the Stirling numbers of two kinds and the Eulerian numbers.