Each positive rational number x > 0 can be written uniquely as x = a/(b − a) for coprime positive integers 0 < a < b. We will identify x with the pair (a, b). In this paper we define for each positive rational x > 0 a simplicial complex Ass(x) = Ass(a, b) called the rational associahedron. It is a pure simplicial complex of dimension a − 2, and its maximal faces are counted by the rational Catalan numberThe cases (a, b) = (n, n + 1) and (a, b) = (n, kn + 1) recover the classical associahedron and its "Fuss-Catalan" generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. We prove that Ass(a, b) is shellable and give nice product formulas for its h-vector (the rational Narayana numbers) and f -vector (the rational Kirkman numbers). We define Ass(a, b) via rational Dyck paths: lattice paths from (0, 0) to (b, a) staying above the line y = a b x. We also use rational Dyck paths to define a rational generalization of noncrossing perfect matchings of [2n]. In the case (a, b) = (n, mn + 1), our construction produces the noncrossing partitions of [(m + 1)n] in which each block has size m + 1.Date: May 2013.
Rowmotion is a simple cyclic action on the distributive lattice of order ideals of a poset: it sends the order ideal x to the order ideal generated by the minimal elements not in x. It can also be computed in ‘slow motion’ as a sequence of local moves. We use the setting of trim lattices to generalize both definitions of rowmotion, proving many structural results along the way. We introduce a flag simplicial complex (similar to the canonical join complex of a semidistributive lattice) and relate our results to recent work of Barnard by proving that extremal semidistributive lattices are trim. As a corollary, we prove that if A is a representation finite algebra and modA has no cycles, then the torsion classes of A ordered by inclusion form a trim lattice.
PURPOSE Thymomas are epithelial neoplasms that represent the most common thymic tumors in adults. These tumors have been shown to harbor a relatively low mutational burden. As a result, there is a lack of genetic alterations that may be used prognostically or targeted therapeutically for this disease. Here, we describe a recurrent gene rearrangement in type B2 + B3 thymomas. PATIENTS AND METHODS A single index case of thymoma was evaluated by an RNA-based solid fusion assay. Separately, tissues from 255,008 unique advanced cancers, including 242 thymomas, were sequenced by hybrid capture–based next-generation DNA sequencing/comprehensive genomic profiling of 186 to 406 genes, including lysine methyltransferase 2A ( KMT2A) rearrangements, and a portion were evaluated for RNA of 265 genes. We characterized molecular and clinicopathologic features of the pertinent fusion-positive patient cases. RESULTS We identified 11 patients with thymomas harboring a gene fusion of KMT2A and mastermind-like transcriptional coactivator 2 ( MAML2). Fusion breakpoints were identified between exon 8, 9, 10, or 11 of KMT2A and exon 2 of MAML2. Fifty-five percent were men, with a median age of 48 years at surgery (range, 29-69 years). Concurrent genomic alterations were infrequent. The 11 thymomas were of B2 or B3 type histology, with 1 case showing foci of thymic carcinoma. The frequency of KMT2A- MAML2 fusion was 4% of all thymomas (10 of 242) and 6% of thymomas of B2 or B3 histology (10 of 169). CONCLUSION KMT2A- MAML2 represents the first recurrent fusion described in type B thymoma. The fusion seems to be specific to type B2 and B3 thymomas, the most aggressive histologic subtypes. The identification of this fusion offers insights into the biology of thymoma and may have clinical relevance for patients with disease refractory to conventional therapeutic modalities.
We say two posets are doppelgängers if they have the same number of P -partitions of each height k. We give a uniform framework for bijective proofs that posets are doppelgängers by synthesizing K-theoretic Schubert calculus techniques of H. Thomas and A. Yong with M. Haiman's rectification bijection and an observation of R. Proctor. Geometrically, these bijections reflect the rational equivalence of certain subvarieties of minuscule flag manifolds. As a special case, we provide the first bijective proof of a 1983 theorem of R. Proctor-that plane partitions of height k in a rectangle are equinumerous with plane partitions of height k in a shifted trapezoid. arXiv:1602.05535v3 [math.CO]
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