scholarly journals Generalized even and odd totally positive matrices

2011 ◽  
Vol 435 (11) ◽  
pp. 2722-2730
Author(s):  
O.Y. Kushel ◽  
P. Sharma
Keyword(s):  
Totally Positive ◽  
Positive Matrices ◽  
Totally Positive Matrices

scholarly journals Sequence Characterization of 3-Dimensional Riordan Arrays and Some Application

2019 ◽  
Vol 74 (4) ◽  
Author(s):  
Roksana Słowik
Keyword(s):  
Sequence Characterization ◽  
3 Dimensional ◽  
Riordan Arrays ◽  
Totally Positive ◽  
Positive Matrices ◽  
Totally Positive Matrices

Abstract We propose the characterization of 3-dimensional Riordan arrays with use of three sequences that is analogous to the representation of 2-dimensional Riordan arrays with use of A and Z-sequence. We also suggest an application of this representation for finding totally positive matrices.

scholarly journals Arrangements of equal minors in the positive Grassmannian

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Miriam Farber ◽  
Alexander Postnikov
Keyword(s):  
Cluster Algebra ◽  
The Other ◽  
Eulerian Number ◽  
Other Hand ◽  
Totally Positive ◽  
Positive Matrices ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets ◽  
Totally Positive Matrices

International audience We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with <i>sorted sets</i>, which earlier appeared in the context of <i>alcoved polytopes</i> and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the <i>Eulerian number</i>. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the <i>weakly separated sets</i>. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated <i>cluster algebra</i>.

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