arXiv subjects

Huan Xiong

Publications and source records attributed to Huan Xiong.

52 records · Page 3Linked to original sources

arXiv2020

Learning to Learn with Variational Information Bottleneck for Domain Generalization

Domain generalization models learn to generalize to previously unseen domains, but suffer from prediction uncertainty and domain shift. In this paper, we address both problems. We introduce a probabilistic meta-learning model for domain generalization, in which classifier parameters shared across domains are modeled as distributions. This enables better handling of prediction uncertainty on unseen domains. To deal with domain shift, we learn domain-invariant representations by the proposed principle of meta variational information bottleneck, we call MetaVIB. MetaVIB is derived from novel variational bounds of mutual information, by leveraging the meta-learning setting of domain generalization. Through episodic training, MetaVIB learns to gradually narrow domain gaps to establish domain-invariant representations, while simultaneously maximizing prediction accuracy. We conduct experiments on three benchmarks for cross-domain visual recognition. Comprehensive ablation studies validate the benefits of MetaVIB for domain generalization. The comparison results demonstrate our method outperforms previous approaches consistently.

cs.CV↗

arXiv2019

Skew doubled shifted plane partitions: calculus and asymptotics

Plane partitions have been widely studied in Mathematics since MacMahon. See, for example, the works by Andrews, Macdonald, Stanley, Sagan and Krattenthaler. The Schur process approach, introduced by Okounkov and Reshetikhin, and further developed by Borodin, Corwin, Corteel, Savelief and Vuletić, has been proved to be a powerful tool in the study of various kinds of plane partitions. The exact enumerations of ordinary plane partitions, shifted plane partitions and cylindric partitions could be derived from two summation formulas for Schur processes, namely, the open summation formula and the cylindric summation formula. In this paper, we establish a new summation formula for Schur processes, called the complete summation formula. As an application, we obtain the generating function and the asymptotic formula for the number of doubled shifted plane partitions, which can be viewed as plane partitions `shifted at the two sides'. We prove that the order of the asymptotic formula depends only on the diagonal width of the doubled shifted plane partition, not on the profile (the skew zone) itself. By using the same methods, the generating function and the asymptotic formula for the number of symmetric cylindric partitions are also derived.

math.CO↗

arXiv2019

Monotonicity properties for ranks of overpartitions

The rank of partitions play an important role in the combinatorial interpretations of several Ramanujan's famous congruence formulas. In 2005 and 2008, the $D$-rank and $M_2$-rank of an overpartition were introduced by Lovejoy, respectively. Let $\overline{N}(m,n)$ and $\overline{N2}(m,n)$ denote the number of overpartitions of $n$ with $D$-rank $m$ and $M_2$-rank $m$, respectively. In 2014, Chan and Mao proposed a conjecture on monotonicity properties of $\overline{N}(m,n)$ and $\overline{N2}(m,n)$. In this paper, we prove the Chan-Mao monotonicity conjecture. To be specific, we show that for any integer $m$ and nonnegative integer $n$, $\overline{N2}(m,n)\leq \overline{N2}(m,n+1)$; and for $(m,n)\neq (0,4)$ with $n\neq\, |m| +2$, we have $\overline{N}(m,n)\leq \overline{N}(m,n+1)$. Furthermore, when $m$ increases, we prove that $\overline{N}(m,n)\geq \overline{N}(m+2,n)$ and $\overline{N2}(m,n)\geq \overline{N2}(m+2,n)$ for any $m,n\geq 0$, which is an analogue of Chan and Mao's result for partitions.

math.CO↗

arXiv2019

On the polynomiality and asymptotics of moments of sizes for random $(n, dn\pm 1)$-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 and Nath-Sellers 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 $\mathbb{E} [X_{n,n+1}^k]$ and $\mathbb{E} [X_{2n+1,2n+3}^k]$ were given by Zaleski and Zeilberger when $k$ is small. Zaleski also studied the expectation and higher moments of $X_{n,dn-1}$ and conjectured some polynomiality properties concerning them in arXiv: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 the beta sets of core partitions. In particular, we show that these $k$-th moments are asymptotically some polynomials of n with degrees at most $2k$, when $d$ is given and $n$ tends to infinity. Moreover, when $d=1$, we derive that the $k$-th moment $\mathbb{E} [X_{n,n+1}^k]$ of $X_{n,n+1}$ is asymptotically equal to $\left(n^2/10\right)^k$ when $n$ tends to infinity. The explicit formulas for the expectations $\mathbb{E} [X_{n,dn+1}]$ and $\mathbb{E} [X_{n,dn-1}]$ are also given. The $(n,dn-1)$-core case in our results proves several conjectures of Zaleski on the polynomiality of the expectation and higher moments of $X_{n,dn-1}$.

math.CO↗

arXiv2018

Difference operators for partitions and some applications

Motivated by the Nekrasov-Okounkov formula on hook lengths, the first author conjectured that the Plancherel average of the $2k$-th power sum of hook lengths of partitions with size $n$ is always a polynomial of $n$ for any $k\in \mathbb{N}$. This conjecture was generalized and proved by Stanley (Ramanujan J., 23(1--3): 91--105, 2010). In this paper, inspired by the work of Stanley and Olshanski on the differential poset of Young lattice, we study the properties of two kinds of difference operators $D$ and $D^-$ defined on functions of partitions. Even though the calculations for higher orders of $D$ are extremely complex, we prove that several well-known families of functions of partitions are annihilated by a power of the difference operator $D$. As an application, our results lead to several generalizations of classic results on partitions, including the marked hook formula, Stanley Theorem, Okada-Panova hook length formula, and Fujii-Kanno-Moriyama-Okada content formula. We insist that the Okada constants $K_r$ arise directly from the computation for a single partition $λ$, without the summation ranging over all partitions of size~$n$.

math.CO↗

arXiv2017

On the largest sizes of certain simultaneous core partitions with distinct parts

Motivated by Amdeberhan's conjecture on $(t,t+1)$-core partitions with distinct parts, various results on the numbers, the largest sizes and the average sizes of simultaneous core partitions with distinct parts were obtained by many mathematicians recently. In this paper, we derive the largest sizes of $(t,mt\pm 1)$-core partitions with distinct parts, which verifies a generalization of Amdeberhan's conjecture. We also prove that the numbers of such partitions with the largest sizes are at most $2$.

math.CO↗

arXiv2017

Polynomiality of certain average weights for oscillating tableaux

We prove that a family of average weights for oscillating tableaux are polynomials in two variables, namely, the length of the oscillating tableau and the size of the ending partition, which generalizes a result of Hopkins and Zhang. Several explicit and asymptotic formulas for the average weights are also derived.

math.CO↗

arXiv2017

Some useful theorems for asymptotic formulas and their applications to skew plane partitions and cylindric partitions

Inspired by the works of Dewar, Murty and Kotěšovec, we establish some useful theorems for asymptotic formulas. As an application, we obtain asymptotic formulas for the numbers of skew plane partitions and cylindric partitions. We prove that the order of the asymptotic formula for the skew plane partitions of fixed width depends only on the width of the region, not on the profile (the skew zone) itself, while this is not true for cylindric partitions.

math.CO↗

arXiv2017

Finite Groups Whose Character Graphs Associated with Codegrees Have No Triangles

Motivated by the Problem $164$ proposed by Y. Berkovich and E. Zhmud' in their book "Characters of finite groups, Part $1$", we give a characterization of finite groups whose irreducible character codegrees are prime powers. This is based on a new kind of character graphs of finite groups associated with codegrees. Such graphs have close and obvious connections with character coedgree graphs. For example, they have the same number of connected components. By analogy with the work of finite groups whose character graphs (associated with degrees) have no triangles, we conduct a result of classifying finite groups whose character graphs associated with codegrees have no triangles in the latter part of this paper.

math.GR↗

arXiv2017

Core partitions with distinct parts

Simultaneous core partitions have attracted much attention since Anderson's work on the number of $(t_1,t_2)$-core partitions. In this paper we focus on simultaneous core partitions with distinct parts. The generating function of $t$-core partitions with distinct parts is obtained. We also prove the results on the number, the largest size and the average size of $(t, t + 1)$-core partitions. This gives a complete answer to a conjecture of Amdeberhan, which is partly and independently proved by Straub, Nath and Sellers, and Zaleski recently.

math.CO↗

arXiv2017

Difference operators for partitions under the Littlewood decomposition

The concept of $t$-difference operator for functions of partitions is introduced to prove a generalization of Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our extension uses a generalization of the notion of Plancherel measure, based on walks in the Young lattice with steps given by the addition of $t$-hooks. It is well-known that the hook lengths of multiples of $t$ can be characterized by the Littlewood decomposition. Our study gives some further information on the contents and hook lengths of other congruence classes modulo $t$.

math.CO↗

arXiv2016

New hook-content formulas for strict partitions

We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the hook length and content statistics. As an application, several new hook-content formulas for strict partitions are derived.

math.CO↗

arXiv2016

Polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions

In 2009, the first author proved the Nekrasov-Okounkov formula on hook lengths for integer partitions by using an identity of Macdonald in the framework of type $\widetilde A$ affine root systems, and conjectured that some summations over the set of all partitions of size $n$ are always polynomials in $n$. This conjecture was generalized and proved by Stanley. Recently, Pétréolle derived two Nekrasov-Okounkov type formulas for $\widetilde C$ and $\widetilde C\,\check{}$ which involve doubled distinct and self-conjugate partitions. Inspired by all those previous works, we establish the polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions.

math.CO↗

arXiv2015

On the largest size of $(t,t+1,..., t+p)$-core partitions

In this paper we prove that Amdeberhan's conjecture on the largest size of $(t, t+1, t+2)$-core partitions is true. We also show that the number of $(t, t + 1, t + 2)$-core partitions with the largest size is $1$ or $2$ based on the parity of $t$. More generally, the largest size of $(t,t+1,..., t+p)$-core partitions and the number of such partitions with the largest size are determined.

math.CO↗

arXiv2015

Partitions with the same hook multiset

It is well-known that two conjugate partitions have the same hook multiset. But two different partitions with the same hook multiset may not be conjugate to each other. In $1977$, Herman and Chung proposed the following question: What are the necessary and sufficient conditions for partitions to be determined by their hook multisets up to conjugation? In this paper, we will answer this question by giving a criterion to determine whether two different partitions with the same hook multiset are conjugate to each other.

math.CO↗

arXiv2014

The number of simultaneous core partitions

Amdeberhan conjectured that the number of $(t,t+1, t+2)$-core partitions is $\sum_{0\leq k\leq [\frac{t}{2}]}\frac{1}{k+1}\binom{t}{2k}\binom{2k}{k}$. In this paper, we obtain the generating function of the numbers $f_t$ of $(t, t + 1, ..., t + p)$-core partitions. In particular, this verifies that Amdeberhan's conjecture is true. We also prove that the number of $(t_1,t_2,..., t_m)$-core partitions is finite if and only if gcd$(t_1,t_2,..., t_m)=1,$ which extends Anderson's result on the finiteness of the number of $(t_1,t_2)$-core partitions for coprime positive integers $t_1$ and $t_2$ and thus rediscover a result of Keith and Nath with a different proof.

math.CO↗