Abstract

A sequence of real numbers {an}n=o,i,2,... is said to be a Polya frequency sequence of or-

der r (or, a PFr sequence, for short) if, for all 1 t r, the determinants of all the

minors of order t of the infinite matrix (aj-^t^o.i^,... (where a^ == 0 if k 0) are nonneg-

ative. So PF2 = log-concave and PF = PF2 = unimodal. It is known that a sequence

{a0, a i , . . . , a^, 0,0,...} is PF if and only if the polynomial X^=0 a,iX% has nonnegative coef-

ficients and only real zeros.

Many sequences in combinatorics are known to be unimodal, PF2 or PF. In this work we

develop general methods which generate new unimodal, PF2 or PF sequences from existing

ones, and we apply such methods to prove the unimodality, PF2 or PF property of sequences

of combinatorial interest.

1 Key-words and phrases: unimodal, log-concave, Polya frequency sequence, total positivity, generat-

ing function, symmetric function, directed graph, poset, labeling, labeled poset, order preserving map, order

polynomial, lattice, distributive lattice.