2,976 research outputs found
Sweep maps: A continuous family of sorting algorithms
We define a family of maps on lattice paths, called sweep maps, that assign
levels to each step in the path and sort steps according to their level.
Surprisingly, although sweep maps act by sorting, they appear to be bijective
in general. The sweep maps give concise combinatorial formulas for the
q,t-Catalan numbers, the higher q,t-Catalan numbers, the q,t-square numbers,
and many more general polynomials connected to the nabla operator and rational
Catalan combinatorics. We prove that many algorithms that have appeared in the
literature (including maps studied by Andrews, Egge, Gorsky, Haglund, Hanusa,
Jones, Killpatrick, Krattenthaler, Kremer, Orsina, Mazin, Papi, Vaille, and the
present authors) are all special cases of the sweep maps or their inverses. The
sweep maps provide a very simple unifying framework for understanding all of
these algorithms. We explain how inversion of the sweep map (which is an open
problem in general) can be solved in known special cases by finding a "bounce
path" for the lattice paths under consideration. We also define a generalized
sweep map acting on words over arbitrary alphabets with arbitrary weights,
which is also conjectured to be bijective.Comment: 21 pages; full version of FPSAC 2014 extended abstrac
Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials
We introduce explicit combinatorial interpretations for the coefficients in
some of the transition matrices relating to skew Hall-Littlewood polynomials
P_lambda/mu(x;t) and Hivert's quasisymmetric Hall-Littlewood polynomials
G_gamma(x;t). More specifically, we provide: 1) the G-expansions of the
Hall-Littlewood polynomials P_lambda, the monomial quasisymmetric polynomials
M_alpha, the quasisymmetric Schur polynomials S_alpha, and the peak
quasisymmetric functions K_alpha; 2) an expansion of P_lambda/mu in terms of
the F_alpha's. The F-expansion of P_lambda/mu is facilitated by introducing
starred tableaux.Comment: 28 pages; added brief discussion of the Hall-Littlewood Q', typos
corrected, added references in response to referee suggestion
A continuous family of partition statistics equidistributed with length
AbstractThis article investigates a remarkable generalization of the generating function that enumerates partitions by area and number of parts. This generating function is given by the infinite product ∏i⩾11/(1−tqi). We give uncountably many new combinatorial interpretations of this infinite product involving partition statistics that arose originally in the context of Hilbert schemes. We construct explicit bijections proving that all of these statistics are equidistributed with the length statistic on partitions of n. Our bijections employ various combinatorial constructions involving cylindrical lattice paths, Eulerian tours on directed multigraphs, and oriented trees
A rooted variant of Stanley's chromatic symmetric function
Richard Stanley defined the chromatic symmetric function of a graph
and asked whether there are non-isomorphic trees and with . We
study variants of the chromatic symmetric function for rooted graphs, where we
require the root vertex to either use or avoid a specified color. We present
combinatorial identities and recursions satisfied by these rooted chromatic
polynomials, explain their relation to pointed chromatic functions and rooted
-polynomials, and prove three main theorems. First, for all non-empty
connected graphs , Stanley's polynomial is irreducible
in for all large enough . The same result holds
for our rooted variant where the root node must avoid a specified color. We
prove irreducibility by a novel combinatorial application of Eisenstein's
Criterion. Second, we prove the rooted version of Stanley's Conjecture: two
rooted trees are isomorphic as rooted graphs if and only if their rooted
chromatic polynomials are equal. In fact, we prove that a one-variable
specialization of the rooted chromatic polynomial (obtained by setting
, , and for ) already distinguishes rooted
trees. Third, we answer a question of Pawlowski by providing a combinatorial
interpretation of the monomial expansion of pointed chromatic functions.Comment: 21 pages; v2: added a short algebraic proof to Theorem 2 (now Theorem
15), we also answer a question of Pawlowski about monomial expansions; v3:
added additional one-variable specialization results, simplified main proof
Regular mosaic pattern development: A study of the interplay between lateral inhibition, apoptosis and differential adhesion
<p>Abstract</p> <p>Background</p> <p>A significant body of literature is devoted to modeling developmental mechanisms that create patterns within groups of initially equivalent embryonic cells. Although it is clear that these mechanisms do not function in isolation, the timing of and interactions between these mechanisms during embryogenesis is not well known. In this work, a computational approach was taken to understand how lateral inhibition, differential adhesion and programmed cell death can interact to create a mosaic pattern of biologically realistic primary and secondary cells, such as that formed by sensory (primary) and supporting (secondary) cells of the developing chick inner ear epithelium.</p> <p>Results</p> <p>Four different models that interlaced cellular patterning mechanisms in a variety of ways were examined and their output compared to the mosaic of sensory and supporting cells that develops in the chick inner ear sensory epithelium. The results show that: 1) no single patterning mechanism can create a 2-dimensional mosaic pattern of the regularity seen in the chick inner ear; 2) cell death was essential to generate the most regular mosaics, even through extensive cell death has not been reported for the developing basilar papilla; 3) a model that includes an iterative loop of lateral inhibition, programmed cell death and cell rearrangements driven by differential adhesion created mosaics of primary and secondary cells that are more regular than the basilar papilla; 4) this same model was much more robust to changes in homo- and heterotypic cell-cell adhesive differences than models that considered either fewer patterning mechanisms or single rather than iterative use of each mechanism.</p> <p>Conclusion</p> <p>Patterning the embryo requires collaboration between multiple mechanisms that operate iteratively. Interlacing these mechanisms into feedback loops not only refines the output patterns, but also increases the robustness of patterning to varying initial cell states.</p
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Technical Comments: Sea Level History
Bilal U. Haq and his co-workers have completed an important update of the chronology of coastal onlap and eustatic fluctuations in Mesowic and Cenowic time. Seismic stratigraphic results are augmented in the new charts by outcrop and well-log studies to document an impressive total of 119 sea level cycles since the beginning of the Triassic. In addition, the Cretaceous results have been published officially for the first time. However, apart from distinguishing between relative changes of coastal onlap and eustasy, the methodology and assumptions are much the same as those used to construct the first version of the "sea level curve" in 1977. In a recent evaluation of the seismic stratigraphic record of sea level change, we drew attention to two problems in particular. 1) All of the observed depositional cycles are assumed by Haq et al. to be eustatic. 2) The global onlap chart, which forms the basis for the smoothed eustatic curve, has little physical meaning
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