Mark Dukes
University of Strathclyde
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Featured researches published by Mark Dukes.
European Journal of Combinatorics | 2010
Fan R. K. Chung; Anders Claesson; Mark Dukes; Ronald L. Graham
Motivated by juggling sequences and bubble sort, we examine permutations on the set {1,2,...,n} with d descents and maximum drop size k. We give explicit formulas for enumerating such permutations for given integers k and d. We also derive the related generating functions and prove unimodality and symmetry of the coefficients.
Journal of High Energy Physics | 2014
Mark Dukes; Einan Gardi; Heather McAslan; Darren J. Scott; Chris D. White
A bstractThe non-Abelian exponentiation theorem has recently been generalised to correlators of multiple Wilson line operators. The perturbative expansions of these correlators exponentiate in terms of sets of diagrams called webs, which together give rise to colour factors corresponding to connected graphs. The colour and kinematic degrees of freedom of individual diagrams in a web are entangled by mixing matrices of purely combinatorial origin. In this paper we relate the combinatorial study of these matrices to properties of partially ordered sets (posets), and hence obtain explicit solutions for certain families of web-mixing matrix, at arbitrary order in perturbation theory. We also provide a general expression for the rank of a general class of mixing matrices, which governs the number of independent colour factors arising from such webs. Finally, we use the poset language to examine a previously conjectured sum rule for the columns of web-mixing matrices which governs the cancellation of the leading subdivergences between diagrams in the web. Our results, when combined with parallel developments in the evaluation of kinematic integrals, offer new insights into the all-order structure of infrared singularities in non-Abelian gauge theories.
Journal of Combinatorial Theory | 2014
Jean-Christophe Aval; Michele D'Adderio; Mark Dukes; Angela Hicks; Yvan Le Borgne
We study the statistics area, bounce and dinv on the set of parallelogram polyominoes having a rectangular m times n bounding box. We show that the bi-statistics (area,bounce) and (area,dinv) give rise to the same q,t-analogue of Narayana numbers which was introduced by two of the authors in [4]. We prove the main conjectures of that paper: the q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is accomplished by providing a symmetric functions interpretation of the q,t-Narayana polynomials which relates them to the famous diagonal harmonics.
Journal of Combinatorial Theory | 2011
Anders Claesson; Mark Dukes; Martina Kubitzke
This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another. We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel. We show that composition matrices on X are in one-to-one correspondence with (2+2)-free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on {1,...,n}.
BMC Systems Biology | 2016
Tom S. Weber; Mark Dukes; Denise C. Miles; Stefan P. Glaser; Shalin H. Naik; Ken R. Duffy
BackgroundCellular barcoding is a recently developed biotechnology tool that enables the familial identification of progeny of individual cells in vivo. In immunology, it has been used to track the burst-sizes of multiple distinct responding T cells over several adaptive immune responses. In the study of hematopoiesis, it revealed fate heterogeneity amongst phenotypically identical multipotent cells. Most existing approaches rely on ex vivo viral transduction of cells with barcodes followed by adoptive transfer into an animal, which works well for some systems, but precludes barcoding cells in their native environment such as those inside solid tissues.ResultsWith a view to overcoming this limitation, we propose a new design for a genetic barcoding construct based on the Cre Lox system that induces randomly created stable barcodes in cells in situ by exploiting inherent sequence distance constraints during site-specific recombination. We identify the cassette whose provably maximal code diversity is several orders of magnitude higher than what is attainable with previously considered Cre Lox barcoding approaches, exceeding the number of lymphocytes or hematopoietic progenitor cells in mice.ConclusionsIts high diversity and in situ applicability, make the proposed Cre Lox based tagging system suitable for whole tissue or even whole animal barcoding. Moreover, it can be built using established technology.
Electronic Notes in Discrete Mathematics | 2016
Mark Dukes; Thomas Selig
The recurrent states of the Abelian sandpile model (ASM) are those states that appear infinitely often. For this reason they occupy a central position in ASM research. We present several new results for classifying recurrent states of the Abelian sandpile model on graphs that may be decomposed in a variety of ways. These results allow us to classify, for certain families of graphs, recurrent states in terms of the recurrent states of its components. We use these decompositions to give recurrence relations for the generating functions of the level statistic on the recurrent configurations. We also interpret our results with respect to the sandpile group.
Journal of Combinatorial Theory | 2010
Mireille Bousquet-Mélou; Anders Claesson; Mark Dukes; Sergey Kitaev
Electronic Journal of Combinatorics | 2010
Mark Dukes; Robert Parviainen
The Journal of Combinatorics | 2011
Mark Dukes; Sergey Kitaev; Jeffrey B. Remmel; Einar Steingrimsson
Journal of Combinatorial Theory | 2013
Mark Dukes; Einan Gardi; Einar Steingrimsson; Chris D. White