Division and the Giambelli Identity

Given two polynomials f(x) and g(x), we extend the formula expressing the remainder in terms of the roots of these two polynomials to the case where f(x) is a Laurent polynomial. This allows us to give new expressions of a Schur function, which generalize the Giambelli identity.Comment: 9 pages, 1 figur

The Alternative Choice of Constitutive Exons throughout Evolution

Alternative cassette exons are known to originate from two processes exonization of intronic sequences and exon shuffling. Herein, we suggest an additional mechanism by which constitutively spliced exons become alternative cassette exons during evolution. We compiled a dataset of orthologous exons from human and mouse that are constitutively spliced in one species but alternatively spliced in the other. Examination of these exons suggests that the common ancestors were constitutively spliced. We show that relaxation of the 59 splice site during evolution is one of the molecular mechanisms by which exons shift from constitutive to alternative splicing. This shift is associated with the fixation of exonic splicing regulatory sequences (ESRs) that are essential for exon definition and control the inclusion level only after the transition to alternative splicing. The effect of each ESR on splicing and the combinatorial effects between two ESRs are conserved from fish to human. Our results uncover an evolutionary pathway that increases transcriptome diversity by shifting exons from constitutive to alternative splicin

Analytic approach for reflected Brownian motion in the quadrant

Random walks in the quarter plane are an important object both of combinatorics and probability theory. Of particular interest for their study, there is an analytic approach initiated by Fayolle, Iasnogorodski and Malyshev, and further developed by the last two authors of this note. The outcomes of this method are explicit expressions for the generating functions of interest, asymptotic analysis of their coefficients, etc. Although there is an important literature on reflected Brownian motion in the quarter plane (the continuous counterpart of quadrant random walks), an analogue of the analytic approach has not been fully developed to that context. The aim of this note is twofold: it is first an extended abstract of two recent articles of the authors of this paper, which propose such an approach; we further compare various aspects of the discrete and continuous analytic approaches.Comment: 19 pages, 5 figures. Extended abstract of the papers arXiv:1602.03054 and arXiv:1604.02918, to appear in Proceedings of the 27th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, Krakow, Poland, 4-8 July 2016 arXiv admin note: text overlap with arXiv:1602.0305

Partial duality of hypermaps

We introduce a collection of new operations on hypermaps, partial duality, which include the classical Euler-Poincar\'e dualities as particular cases. These operations generalize the partial duality for maps, or ribbon graphs, recently discovered in a connection with knot theory. Partial duality is different from previous studied operations of S. Wilson, G. Jones, L. James, and A. Vince. Combinatorially hypermaps may be described in one of three ways: as three involutions on the set of flags ($\tau$-model), or as three permutations on the set of half-edges ($\sigma$-model in orientable case), or as edge 3-colored graphs. We express partial duality in each of these models.Comment: 19 pages, 16 figure