21 research outputs found

    A refinement of the Shuffle Conjecture with cars of two sizes and t=1/qt=1/q

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    The original Shuffle Conjecture of Haglund et al. has a symmetric function side and a combinatorial side. The symmetric function side may be simply expressed as where \nabla is the Macdonald polynomial eigen-operator of Bergeron and Garsia and hμh_\mu is the homogeneous basis indexed by μ=(μ1,μ2,...,μk)\mu=(\mu_1,\mu_2,...,\mu_k) partitions of n. The combinatorial side q,t-enumerates a family of Parking Functions whose reading word is a shuffle of k successive segments of 1,2,3,...,n of respective lengths μ1,μ2,...,μk\mu_1,\mu_2,...,\mu_k. It can be shown that for t=1/q the symmetric function side reduces to a product of q-binomial coefficients and powers of q. This reduction suggests a surprising combinatorial refinement of the general Shuffle Conjecture. Here we prove this refinement for k=2 and t=1/q. The resulting formula gives a q-analogue of the well studied Narayana numbers.Comment: 17 pages, 11 figure

    Some remarkable new Plethystic Operators in the Theory of Macdonald Polynomials

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    In the 90's a collection of Plethystic operators were introduced in [3], [7] and [8] to solve some Representation Theoretical problems arising from the Theory of Macdonald polynomials. This collection was enriched in the research that led to the results which appeared in [5], [6] and [9]. However since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished. As a consequence of very recent publications [4], [11], [19], [20], [21], a truly remarkable expansion of this theory has taken place. However most of this work has appeared in a language that is virtually inaccessible to practitioners of Algebraic Combinatorics. Yet, these developments have led to a variety of new conjectures in [2] in the Combinatorics and Symmetric function Theory of Macdonald Polynomials. The present work results from an effort to obtain in an elementary and accessible manner all the background necessary to construct the symmetric function side of some of these new conjectures. It turns out that the above mentioned unpublished results provide precisely the tools needed to carry out this project to its completion

    Two special cases of the Rational Shuffle Conjecture

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    The Classical Shuffle Conjecture of Haglund et al. (2005) has a symmetric function side and a combinatorial side. The combinatorial side q,tq,t-enumerates parking functions in the n×nn ×n lattice. The symmetric function side may be simply expressed as en∇ e_n , where is the Macdonald eigen-operator introduced by Bergeron and Garsia (1999) and ene_n is the elementary symmetric function. The combinatorial side has been extended to parking functions in the m×nm ×n lattice for coprime m,nm,n by Hikita (2012). Recently, Gorsky and Negut have been able to extend the Shuffle Conjecture by combining their work (2012a, 2012b, 2013) (related to work of Schiffmann and Vasserot (2011, 2013)) with Hikita's combinatorial results. We prove this new conjecture for the cases m=2m=2 and n=2n=2

    Recall patterns and risk of primary liver cancer for subcentimeter ultrasound liver observations: a multicenter study

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    BACKGROUND: Patients with cirrhosis and subcentimeter lesions on liver ultrasound are recommended to undergo short-interval follow-up ultrasound because of the presumed low risk of primary liver cancer (PLC). AIMS: The aim of this study is to characterize recall patterns and risk of PLC in patients with subcentimeter liver lesions on ultrasound. METHODS: We conducted a multicenter retrospective cohort study among patients with cirrhosis or chronic hepatitis B infection who had subcentimeter ultrasound lesions between January 2017 and December 2019. We excluded patients with a history of PLC or concomitant lesions ≥1 cm in diameter. We used Kaplan Meier and multivariable Cox regression analyses to characterize time-to-PLC and factors associated with PLC, respectively. RESULTS: Of 746 eligible patients, most (66.0%) had a single observation, and the median diameter was 0.7 cm (interquartile range: 0.5-0.8 cm). Recall strategies varied, with only 27.8% of patients undergoing guideline-concordant ultrasound within 3-6 months. Over a median follow-up of 26 months, 42 patients developed PLC (39 HCC and 3 cholangiocarcinoma), yielding an incidence of 25.7 cases (95% CI, 6.2-47.0) per 1000 person-years, with 3.9% and 6.7% developing PLC at 2 and 3 years, respectively. Factors associated with time-to-PLC were baseline alpha-fetoprotein \u3e10 ng/mL (HR: 4.01, 95% CI, 1.85-8.71), platelet count ≤150 (HR: 4.90, 95% CI, 1.95-12.28), and Child-Pugh B cirrhosis (vs. Child-Pugh A: HR: 2.54, 95% CI, 1.27-5.08). CONCLUSIONS: Recall patterns for patients with subcentimeter liver lesions on ultrasound varied widely. The low risk of PLC in these patients supports short-interval ultrasound in 3-6 months, although diagnostic CT/MRI may be warranted for high-risk subgroups such as those with elevated alpha-fetoprotein levels

    Two special cases of the Rational Shuffle Conjecture

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    The Classical Shuffle Conjecture of Haglund et al. (2005) has a symmetric function side and a combinatorial side. The combinatorial side q,tq,t-enumerates parking functions in the n×nn ×n lattice. The symmetric function side may be simply expressed as en∇ e_n , where is the Macdonald eigen-operator introduced by Bergeron and Garsia (1999) and ene_n is the elementary symmetric function. The combinatorial side has been extended to parking functions in the m×nm ×n lattice for coprime m,nm,n by Hikita (2012). Recently, Gorsky and Negut have been able to extend the Shuffle Conjecture by combining their work (2012a, 2012b, 2013) (related to work of Schiffmann and Vasserot (2011, 2013)) with Hikita's combinatorial results. We prove this new conjecture for the cases m=2m=2 and n=2n=2

    Two special cases of the Rational Shuffle Conjecture

    No full text
    Abstract. The Classical Shuffle Conjecture of Haglund et al. (2005) has a symmetric function side and a combinatorial side. The combinatorial side q, t-enumerates parking functions in the n × n lattice. The symmetric function side may be simply expressed as ∇en, where ∇ is the Macdonald eigen-operator introduced by Bergeron and Garsia (1999) and en is the elementary symmetric function. The combinatorial side has been extended to parking functions in the m × n lattice for coprime m, n by Hikita (2012). Recently, Gorsky and Negut have been able to extend the Shuffle Conjecture by combining their work (2012a, 2012b, 2013) (related to work of Schiffmann and Vasserot (2011, 2013)) with Hikita’s combinatorial results. We prove this new conjecture for the cases m = 2 and n = 2. Résumé. La Conjecture “Shuffle ” Classique de Haglund et al. (2005) présente un côté fonctions symétriques et un côté combinatoire. Le côté combinatoire q, t-énumère les fonctions parking dans le n × n treillis. Le côt
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