Arithmetic properties of ℓ-regular partitions

For a given prime p, by studying p  -dissection identities for Ramanujan?s theta functions ψ(q)$\psi (q)$ and f(−q)$f(-q)$, we derive infinite families of congruences modulo 2 for some ?  -regular partition functions, where ?=2,4,5,8,13,16$?=2,4,5,8,13,16$.     

Column-to-row operations on partitions: Garden of Eden partitions

Conjugation and the Bulgarian solitaire move are the extreme cases of general column-to-row operations on integer partitions. Each operation generates a state diagram on the partitions of n. Garden of Eden states are those with no preimage under the operation in question. In this note, we determine the number of Garden of Eden partitions for all n and column-to-row operations.     

On the polynomiality and asymptotics of moments of sizes for random(n,dn ± 1)-core partitions with distinct parts

Amdeberhan’s conjectures on the enumeration,the average size,and the largest size of(n,n+1)-core partitions with distinct parts have motivated many research on this topic.Recently,Straub(2016)and Nath and Sellers(2017)obtained formulas for the numbers of(n,dn-1)-and(n,dn+1)-core partitions with distinct parts,respectively.Let X_s,t be the size of a uniform random(s,t)-core partition with distinct parts when s and t are coprime to each other.Some explicit formulas for the k-th moments E[X_n,n+1^k]and E[X_(2 n+1,2 n+3)^k]were given by Zaleski and Zeilberger(2017)when k is small.Zaleski(2017)also studied the expectation and higher moments of X_n,dn-1 and conjectured some polynomiality properties concerning them in ar Xiv:1702.05634.Motivated by the above works,we derive several polynomiality results and asymptotic formulas for the k-th moments of X_n,dn+1 and X_n,dn-1 in this paper,by studying theβ-sets of core partitions.In particular,we show that these k-th moments are asymptotically some polynomials of n with degrees at most 2 k,when d is given and n tends to infinity.Moreover,when d=1,we derive that the k-th moment E[X_n,n+1^k]of X_n,n+1 is asymptotically equal to(n²/10)^kwhen n tends to infinity.The explicit formulas for the expectations E[X_n,dn+1]and E[X_n,dn-1]are also given.The(n,dn-1)-core case in our results proves several conjectures of Zaleski(2017)on the polynomiality of the expectation and higher moments of X_n,dn-1.     

Symmetric functions,codes of partitions and the KP hierarchy

We consider an operator of Bernstein for symmetric functions and give an explicit formula for its action on an arbitrary Schur function. This formula is given in a remarkably simple form when written in terms of some notation based on the code of a partition. As an application, we give a new and very simple proof of a classical result for the KP hierarchy, which involves the Plücker relations for Schur function coefficients in a τ-function for the hierarchy. This proof is especially compact because we are able to restate the Plücker relations in a form that is symmetrical in terms of partition code notation.     

Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence

We prove that the theta correspondence for the dual pair , for B an indefinite quaternion algebra over ℚ, acting on modular forms of odd square-free level, preserves rationality and p-integrality in both directions. As a consequence, we deduce the rationality of certain period ratios of modular forms and even p-integrality of these ratios under the assumption that p does not divide a certain L-value. The rationality is applied to give a direct construction of isogenies between new quotients of Jacobians of Shimura curves, completely independent of Faltings’ isogeny theorem. To my parents.     

Exotic smooth structures on small 4-manifolds with odd signatures

Let M be (2n-1)\mathbbCP²#2n[`(\mathbbCP)]<sup>2</sup>(2n-1)\mathbb{CP}^{2}\#2n\overline{\mathbb{CP}}{}^{2} for any integer n≥1. We construct an irreducible symplectic 4-manifold homeomorphic to M and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to M. We also construct such exotic smooth structures when M is \mathbbCP<sup>2</sup>#4[`(\mathbbCP)]²\mathbb{CP}{}^{2}\#4\overline {\mathbb{CP}}{}^{2} or 3\mathbbCP²#k[`(\mathbbCP)]²3\mathbb{CP}{}^{2}\#k\overline{\mathbb{CP}}{}^{2} for k=6,8,10.     

Arithmetic of the 13-regular partition function modulo 3

Let b ₁₃(n) denote the number of 13-regular partitions of n. We study in this paper the behavior of b ₁₃(n) modulo 3 where n≡1 (mod 3). In particular, we identify an infinite family of arithmetic progressions modulo arbitrary powers of 3 such that b ₁₃(n)≡0 (mod 3).     

Connectivity properties of horospheres in Euclidean buildings and applications to finiteness properties of discrete groups

Let G(O_S)\mathbf{G}(\mathcal{O}_{S}) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S)\mathbf{G}(\mathcal{O}_{S}) provided the K-rank of G is 1. This shows that the general upper bound for the finiteness length of G(O_S)\mathbf{G}(\mathcal{O}_{S}) established in an earlier paper is sharp in this case.     

Concentration of the Ratio between the Geometric and Arithmetic Means

We explore the concentration properties of the ratio between the geometric mean and the arithmetic mean, showing that for certain sequences of weights, one does obtain concentration around a value that depends on the sequence.     

?-adic properties of smallest parts functions

We prove explicit congruences modulo powers of arbitrary primes for three smallest parts functions: one for partitions, one for overpartitions, and one for partitions without repeated odd parts. The proofs depend on ?-adic properties of certain modular forms and mock modular forms of weight 3/2 with respect to the Hecke operators T(?2m).     

The Erd?s-Ko-Rado properties of set systems defined by double partitions

Let F be a family of subsets of a finite set V. The star ofFatv∈V is the sub-family {A∈F:v∈A}. We denote the sub-family {A∈F:|A|=r} by F^(r).A double partitionP of a finite set V is a partition of V into large sets that are in turn partitioned into small sets. Given such a partition, the family F(P)induced byP is the family of subsets of V whose intersection with each large set is either contained in just one small set or empty.Our main result is that, if one of the large sets is trivially partitioned (that is, into just one small set) and 2r is not greater than the least cardinality of any maximal set of F(P), then no intersecting sub-family of F(P)^(r) is larger than the largest star of F(P)^(r). We also characterise the cases when every extremal intersecting sub-family of F(P)^(r) is a star of F(P)^(r).     

(−1)-enumeration of plane partitions with complementation symmetry

We compute the weighted enumeration of plane partitions contained in a given box with complementation symmetry where adding one half of an orbit of cubes and removing the other half of the orbit changes the weight by −1 as proposed by Kuperberg in [Electron. J. Combin. 5 (1998) R46, pp. 25, 26]. We use nonintersecting lattice path families to accomplish this for transpose-complementary, cyclically symmetric transpose-complementary and totally symmetric self-complementary plane partitions. For symmetric transpose-complementary and self-complementary plane partitions we get partial results. We also describe Kuperberg's proof for the case of cyclically symmetric self-complementary plane partitions.     

Multigraded combinatorial Hopf algebras and refinements of odd and even subalgebras

We develop a theory of multigraded (i.e., ℕ ^l -graded) combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar et al. (Compos. Math. 142:1–30, 2006). In particular we introduce the notion of canonical k-odd and k-even subalgebras associated with any multigraded combinatorial Hopf algebra, extending simultaneously the work of Aguiar et al. and Ehrenborg. Among our results are specific categorical results for higher level quasisymmetric functions, several basis change formulas, and a generalization of the descents-to-peaks map.     

Log-convexity properties of Schur functions and generalized hypergeometric functions of matrix argument

We establish a positivity property for the difference of products of certain Schur functions, s _λ (x), where λ varies over a fundamental Weyl chamber in ? ⁿ and x belongs to the positive orthant in ? ⁿ . Further, we generalize that result to the difference of certain products of arbitrary numbers of Schur functions. We also derive a log-convexity property of the generalized hypergeometric functions of two Hermitian matrix arguments, and we show how that result may be extended to derive higher-order log-convexity properties.     

Wavelet shrinkage estimators of Calderón-Zygmund operators with odd kernels

Wavelet shrinkage is a strategy to obtain a nonlinear approximation to a given function f and is widely used in data compression,signal processing and statistics,etc.For Calder′on-Zygmund operators T,it is interesting to construct estimator of T f,based on wavelet shrinkage estimator of f.With the help of a representation of operators on wavelets,due to Beylkin et al.,an estimator of T f is presented in this paper.The almost everywhere convergence and norm convergence of the proposed estimators are established.