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Search: MSC category 05C38 ( Paths and cycles [See also 90B10] )

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1. CJM 2005 (vol 57 pp. 82)

Fallat, Shaun M.; Gekhtman, Michael I.
Jordan Structures of Totally Nonnegative Matrices
An $n \times n$ matrix is said to be totally nonnegative if every minor of $A$ is nonnegative. In this paper we completely characterize all possible Jordan canonical forms of irreducible totally nonnegative matrices. Our approach is mostly combinatorial and is based on the study of weighted planar diagrams associated with totally nonnegative matrices.

Keywords:totally nonnegative matrices, planar diagrams,, principal rank, Jordan canonical form
Categories:15A21, 15A48, 05C38

2. CJM 2001 (vol 53 pp. 758)

Goulden, I. P.; Jackson, D. M.; Latour, F. G.
Inequivalent Transitive Factorizations into Transpositions
The question of counting minimal factorizations of permutations into transpositions that act transitively on a set has been studied extensively in the geometrical setting of ramified coverings of the sphere and in the algebraic setting of symmetric functions. It is natural, however, from a combinatorial point of view to ask how such results are affected by counting up to equivalence of factorizations, where two factorizations are equivalent if they differ only by the interchange of adjacent factors that commute. We obtain an explicit and elegant result for the number of such factorizations of permutations with precisely two factors. The approach used is a combinatorial one that rests on two constructions. We believe that this approach, and the combinatorial primitives that have been developed for the ``cut and join'' analysis, will also assist with the general case.

Keywords:transitive, transposition, factorization, commutation, cut-and-join
Categories:05C38, 15A15, 05A15, 15A18

3. CJM 1997 (vol 49 pp. 301)

Merlini, Donatella; Rogers, Douglas G.; Sprugnoli, Renzo; Verri, M. Cecilia
On some alternative characterizations of Riordan arrays
We give several new characterizations of Riordan Arrays, the most important of which is: if $\{d_{n,k}\}_{n,k \in {\bf N}}$ is a lower triangular array whose generic element $d_{n,k}$ linearly depends on the elements in a well-defined though large area of the array, then $\{d_{n,k}\}_{n,k \in {\bf N}}$ is Riordan. We also provide some applications of these characterizations to the lattice path theory.

Categories:05A15, 05C38

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