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.
Previous Page Next Page