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a r t i c l e i n f o a b s t r a c t A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally positive matrix and show interesting connections with incidence problems in… (More)

We introduce n(n − 1)/2 natural involutions (" toggles ") on the set S of non-crossing partitions π of size n, along with certain composite operations obtained by composing these involutions. We show that for many operations T of this kind, a surprisingly large family of functions f on S (including the function that sends π to the number of blocks of π)… (More)

Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Abstract. We consider the set of Hermitian matrices corresponding to a given… (More)

We consider the response matrices in certain weighted networks that display a circular symmetry. It had been observed empirically that these exhibit several paired (multiplicity two) eigenvalues. Here, this pairing is explained analytically for a version of the model more general than the original. The exact number of necessarily paired eigenvalues is given… (More)