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1. Two-sided permutation statistics via symmetric functions

   https://doi.org/10.1017/fms.2024.89  

   Gessel, Ira M; Zhuang, Yan (November 2024, Forum of Mathematics, Sigma)

   Abstract Given a permutation statistic$$\operatorname {\mathrm {st}}$$, define its inverse statistic$$\operatorname {\mathrm {ist}}$$by. We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {ist}}_{2}$$whenever$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {st}}_{2}$$are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {ist}}_{2}$$can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {st}}_{2}$$. Our work leads to a rederivation of Stanley’s generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and$$\gamma $$-positivity. 

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2. A spatial mutation model with increasing mutation rates

   https://doi.org/10.1017/jpr.2022.120  

   Chao, Brian; Schweinsberg, Jason (December 2023, Journal of Applied Probability)

   Abstract We consider a spatial model of cancer in which cells are points on thed-dimensional torus$$\mathcal{T}=[0,L]^d$$, and each cell with$$k-1$$mutations acquires akth mutation at rate$$\mu_k$$. We assume that the mutation rates$$\mu_k$$are increasing, and we find the asymptotic waiting time for the first cell to acquirekmutations as the torus volume tends to infinity. This paper generalizes results on waiting for$$k\geq 3$$mutations in Fooet al.(2020), which considered the case in which all of the mutation rates$$\mu_k$$are the same. In addition, we find the limiting distribution of the spatial distances between mutations for certain values of the mutation rates. 

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3. Quadratic Chabauty for modular curves: algorithms and examples

   https://doi.org/10.1112/S0010437X23007170  

   Balakrishnan, Jennifer S.; Dogra, Netan; Müller, J. Steffen; Tuitman, Jan; Vonk, Jan (June 2023, Compositio Mathematica)

   We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus$$g>1$$whose Jacobians have Mordell–Weil rank$$g$$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients$$X_0^+(N)$$of prime level$$N$$, the curve$$X_{S_4}(13)$$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve$$X_{\scriptstyle \mathrm { ns}} ^+ (17)$$. 

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4. On oriented cycles in randomly perturbed digraphs

   https://doi.org/10.1017/S0963548323000391  

   Araujo, Igor; Balogh, József; Krueger, Robert A; Piga, Simón; Treglown, Andrew (November 2023, Combinatorics, Probability and Computing)

   Abstract In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every$$\alpha \gt 0$$, there exists a constant$$C$$such that for every$$n$$-vertex digraph of minimum semi-degree at least$$\alpha n$$, if one adds$$Cn$$random edges then asymptotically almost surely the resulting digraph contains a consistently oriented Hamilton cycle. We generalize their result, showing that the hypothesis of this theorem actually asymptotically almost surely ensures the existence of every orientation of a cycle of every possible length, simultaneously. Moreover, we prove that we can relax the minimum semi-degree condition to a minimum total degree condition when considering orientations of a cycle that do not contain a large number of vertices of indegree$$1$$. Our proofs make use of a variant of an absorbing method of Montgomery. 

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5. A virtual PGL–SL correspondence for projective surfaces

   https://doi.org/10.1112/mod.2024.13  

   van_Bree, Dirk; Gholampour, Amin; Jiang, Yunfeng; Kool, Martijn (January 2025, Moduli)

   Abstract For a smooth projective surface$$X$$satisfying$$H_1(X,\mathbb{Z}) = 0$$and$$w \in H^2(X,\mu _r)$$, we study deformation invariants of the pair$$(X,w)$$. Choosing a Brauer–Severi variety$$Y$$(or, equivalently, Azumaya algebra$$\mathcal{A}$$) over$$X$$with Stiefel–Whitney class$$w$$, the invariants are defined as virtual intersection numbers on suitable moduli spaces of stable twisted sheaves on$$Y$$constructed by Yoshioka (or, equivalently, moduli spaces of$$\mathcal{A}$$-modules of Hoffmann–Stuhler). We show that the invariants do not depend on the choice of$$Y$$. Using a result of de Jong, we observe that they are deformation invariants of the pair$$(X,w)$$. For surfaces with$$h^{2,0}(X) \gt 0$$, we show that the invariants can often be expressed as virtual intersection numbers on Gieseker–Maruyama–Simpson moduli spaces of stable sheaves on$$X$$. This can be seen as a$${\rm PGL}_r$$–$${\rm SL}_r$$correspondence. As an application, we express$${\rm SU}(r) / \mu _r$$Vafa–Witten invariants of$$X$$in terms of$${\rm SU}(r)$$Vafa–Witten invariants of$$X$$. We also show how formulae from Donaldson theory can be used to obtain upper bounds for the minimal second Chern class of Azumaya algebras on$$X$$with given division algebra at the generic point. 

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