Nonnegative Minors of Minor Matrices

Using the relationship between totally nonnegative matrices and directed acyclic weighted planar networks, we show that 2× 2 minors of minor matrices of totally nonnegative matrices are also nonnegative. We give a combinatorial interpretation for the minors of minor matrices in terms of the weights of families of paths in a network. Introduction By attaching weights to the edges of a finite, directed, acyclic planar network we form the corresponding weight matrix. This weight matrix encodes important information about the network. For the types of networks relevant to this paper, a result of Lindström [5, Lemma 1] shows that these matrices are totally nonnegative, i.e. any minor is a subtraction-free expression in the weights of the network. In this paper we extend Lindström’s argument by showing that 2 × 2 minors of the minor matrices (defined in §1) of the weight matrix are also nonnegative. Moreover, we show that these minors of the minor matrices will be subtraction-free expressions in the weights of the original network. As an application of the main theorem of this paper we give an extension of a conjecture, independently made by McNamara and Sagan [6, Conjecture 7.1] and R. P. Stanley, about infinite log-concavity. To state their conjecture we introduce some of the relevant background. Let {an}n=0 be a sequence of nonnegative real numbers. We say the sequence is log-concave if the new sequence {bn} given by bn = an−an−1an+1 still consists of nonnegative numbers, where a−1 = 0. If every iteration of this procedure creates another nonnegative sequence, then we say that the original sequence is infinitely log-concave. Notice that if a polynomial ∑m i=0 aix i has only real negative roots, then the sequence {an}n=0 (where an = 0 if n > m) is nonnegative. The statement is as follows: Infinite Log-concavity Conjecture. If ∑m i=0 aix i has only real negative roots then the polynomial ∑n i=0(a 2 i−ai−1ai+1)x also has only real negative roots. In particular, the sequence {an} is infinitely log-concave. Petter Brändén [1] recently proved this conjecture, using complex-analytic techniques applied to symmetric polynomials. We were led to our extension (which is stated in §5) by first noticing that the sequence {an} gives rise to a totally nonnegative matrix A and the infinite 2010 Mathematics Subject Classification. Primary 05C21, Secondary 05C22, 05C30, 26C10.