Core Partitions Research Articles

Self-conjugate [formula omitted]-core partitions and free Motzkin paths

Simultaneous core partitions have been extensively exploited after Anderson’s work on the enumeration of (s,t)-core partitions. Ford, Mai and Sze established a bijection between self-conjugate (s,t)-core partitions and lattice paths in the ⌊s2⌋×⌊t2⌋ rectangle consisting of north and east steps, thereby showing that the number of such partitions is given by ⌊s2⌋+⌊t2⌋⌊s2⌋ for relatively prime integers s and t. In this paper, we explore self-conjugate (s,s+d,s+2d)-core partitions in the spirit of the work of Ford, Mai and Sze. We provide a lattice path interpretation of self-conjugate (s,s+d,s+2d)-core partitions in terms of free Motzkin paths and obtain the enumeration of such core partitions.

A new analogue of $t$-core partitions

By analogy with $t$-core partitions, we study $\overline {a}_t(n)$, given by $\sum _{n=0}^{\infty }\overline {a}_t(n)q^n\break ={\phi (-q^t)^t}/{\phi (-q)}.$ We obtain multiplicative formulas and arithmetic properties of $\overline {a}_{t}(n)$ for $t\in \

Keys and Demazure crystals for Kac–Moody algebras

The Key map is an important tool in the determination of the Demazure crystals associated to Kac–Moody algebras. In finite type A, it can be computed in the tableau realization of crystals by a simple combinatorial procedure due to Lascoux and Schützenberger. We show that this procedure is a part of a more general construction holding in the Kac–Moody case that we illustrate in finite types and affine type A. In affine type A, we introduce higher level generalizations of core partitions which are expected to play an important role in the representation theory of Ariki–Koike algebras.

The largest size of an (s,s + 1)-core partition with parts of the same parity

Finding the largest size of a partition under certain restrictions has been an interesting subject to study. For example, it is proved by Olsson and Stanton that for two coprime integers [Formula: see text] and [Formula: see text], the largest size of an [Formula: see text]-core partition is [Formula: see text]. Xiong found a formula for the largest size of a [Formula: see text]-core partitions with distinct parts. In this paper, we find an explicit formula for the largest size of an [Formula: see text]-core partition such that all parts are odd (or even).

The (s,s + d,…,s + pd)-core partitions and the rational Motzkin paths

In this paper, we propose an (s+d,d)-abacus for (s,s+d,…,s+pd)-core partitions and establish a bijection between the (s,s+d,…,s+pd)-core partitions and the rational Motzkin paths of type (s+d,−d). This result not only gives a lattice path interpretation of the (s,s+d,…,s+pd)-core partitions but also counts them with an explicit formula. Also we enumerate (s,s+1,…,s+p)-core partitions with k corners and self-conjugate (s,s+1,…,s+p)-core partitions.

Counting self-conjugate $$(s,s+1,s+2)$$-core partitions

We are concerned with counting self-conjugate $(s,s+1,s+2)$-core partitions. A Motzkin path of length $n$ is a path from $(0,0)$ to $(n,0)$ which stays above the $x$-axis and consists of the up $U=(1,1)$, down $D=(1,-1)$, and flat $F=(1,0)$ steps. We say that a Motzkin path of length $n$ is symmetric if its reflection about the line $x=n/2$ is itself. In this paper, we show that the number of self-conjugate $(s,s+1,s+2)$-cores is equal to the number of symmetric Motzkin paths of length $s$, and give a closed formula for this number.

Simultaneous core partitions with nontrivial common divisor

A tremendous amount of research has been done in the last two decades on (s, t)-core partitions when s and t are relatively prime integers. Here we change perspective slightly and explore properties of (s, t)-core and $$(\bar{s},\bar{t})$$ -core partitions for s and t with a nontrivial common divisor g. We begin by recovering, using the g-core and g-quotient construction, the generating function for (s, t)-core partitions first obtained by Aukerman et al. (Discrete Math 309(9):2712–2720, 2009). Then, using a construction developed by the first two authors, we obtain a generating function for the number of $$(\bar{s},\bar{t})$$ -core partitions of n. Our approach allows for new results on t-cores and self-conjugate t-cores that are notg-cores and $$\bar{t}$$ -cores that are not $$\bar{g}$$ -cores, thus strengthening positivity results of Ono and Granville (Trans Am Soc 348:221–228, 1996), Baldwin et al. (J Algebra 297:438–452, 2006), and Kiming (J Number Theory 60:97–102, 1996). We then move to bijections between bar-core partitions and self-conjugate partitions. We give a new, short proof of a correspondence between self-conjugate t-core and $$\bar{t}$$ -core partitions when t is odd and positive first due to Yang (Ramanujan J 44:197, 2019). Then, using two different lattice-path labelings, one due to Ford et al. (J Number Theory 129:858–865, 2009), the other to Bessenrodt and Olsson (J Algebra 306:3–16, 2006), we give a bijection between self-conjugate (s, t)-core and $$(\bar{s},\bar{t})$$ -core partitions when s and t are odd and coprime. We end this section with a bijection between self-conjugate (s, t)-core and $$(\bar{s},\bar{t})$$ -core partitions when s and t are odd and nontrivial g which uses the results stated above. We end the paper by noting (s, t)-core and $$(\bar{s}, \bar{t})$$ -core partitions inherit Ramanujan-type congruences from those of g-core and $$\bar{g}$$ -core partitions.

Rational Shi tableaux and the skew length statistic

We define two refinements of the skew length statistic on simultaneous core partitions. The first one relies on hook lengths and is used to prove a refined version of the theorem stating that the skew length is invariant under conjugation of the core. The second one is equivalent to a generalisation of Shi tableaux to the rational level of Catalan combinatorics. We prove that the rational Shi tableau is injective. Moreover we present a uniform definition of the rational Shi tableau for Weyl groups and conjecture injectivity in the general case.

The Asymptotic Normality of $(s,s+1)$-Cores with Distinct Parts

Simultaneous core partitions are important objects in algebraic combinatorics. Recently there has been interest in studying the distribution of sizes among all $(s,t)$-cores for coprime $s$ and $t$. Zaleski (2017) gave strong evidence that when we restrict our attention to $(s,s+1)$-cores with distinct parts, the resulting distribution is approximately normal. We prove his conjecture by applying the Combinatorial Central Limit Theorem and mixing the resulting normal distributions.

Bijections between [formula omitted]-core partitions and [formula omitted]-tuples

This note introduces some bijections relating core partitions and tuples of integers. We apply these bijections to count the number of cores with various types of restriction, including fixed number of parts, limited size of parts, parts divisible by some integer, and distinct parts. For example, we prove that the number of 2t-core partitions into l even parts equals the number of t-core partitions into l parts. We also generalize one expression for simultaneous cores, which was given by Baek, Nam and Yu, recently. Subsequently, we use this expression to obtain recurrence satisfied by numbers of (s,s+1,…,s+r)−core partitions for s≥1.

Self-conjugate partitions with the same disparity

We give a bijection between the set of ordinary partitions and that of self-conjugate partitions with some restrictions. Also, we show the relationship between hook lengths of a self-conjugate partition and its corresponding partition via the bijection. As a corollary, we give new combinatorial interpretations for the Catalan number and the Motzkin number in terms of self-conjugate simultaneous core partitions.

On self-conjugate (s,s + 1,…,s + k)-core partitions

Simultaneous core partitions have been widely studied since Anderson's work on the enumeration of (s,t)-core partitions. Amdeberhan and Leven showed that the number of (s,s+1,…,s+k)-core partitions is equal to the number of (s,k)-Dyck paths. In this paper, we prove that self-conjugate (s,s+1,…,s+k)-core partitions are equinumerous with symmetric (s,k)-Dyck paths, confirming a conjecture posed by Cho, Huh and Sohn.

On the number of simultaneous core partitions with [formula omitted]-distinct parts

We investigate the number Nd,r(s) of (s,s+r)-core integer partitions with d-distinct parts. Our first main result is a proof of a recurrence relation conjectured by Sahin in 2018. We also derive generating functions, asymptotics, and exact formulas for Nd,r(s) when r is within d of a multiple of s. Finally, we exhibit a surprising connection to A-restricted compositions.

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 Xs,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 +1 ] and E[X 2+1,2+3 ] were given by Zaleski and Zeilberger (2017) when k is small. Zaleski (2017) also studied the expectation and higher moments of Xn,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 Xn,dn+1 and Xn,dn−1 in this paper, by studying the β-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 E[X +1 ] of Xn,n+1 is asymptotically equal to (n2/10)k when n tends to infinity. The explicit formulas for the expectations E[Xn,dn+1] and E[Xn,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 Xn,dn−1.

Congruence properties of pk(n)

We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see text]) can be expressed as the product of an integral linear combination of modular functions on [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]) for any [Formula: see text] and [Formula: see text]. By investigating the properties of the modular equations of the [Formula: see text]th order under the Atkin [Formula: see text]-operator, we obtain that these generating functions are determined by some linear recurring sequences. Utilizing the periodicity of these linear recurring sequences modulo [Formula: see text], we are led to infinite families of congruences for [Formula: see text] modulo any [Formula: see text] with [Formula: see text] and periodic relations between the values of [Formula: see text] modulo powers of [Formula: see text]. As applications, infinite families of congruences for many partition functions such as [Formula: see text]-core partition functions, the partition function and Andrews’ spt-function are easily obtained.

Bijections between bar-core and self-conjugate core partitions

In the recent decades, core partitions and bar core partitions have attracted much attention from combinatorial researchers. In this paper, we build a bijection between $$\bar{t}$$ -cores and self-conjugate t-cores for odd integers $$t\ge 3$$ . To this end, we first establish a bijection between $$\bar{t}$$ -cores and self-conjugate $$(t-1)$$ -cores for odd integers $$t\ge 3$$ , then we prove that this bijection can be extended to a bijection between self-conjugate $$(t-1)$$ -cores and self-conjugate t-cores for odd integers $$t\ge 3$$ .

The existence of a unique core partition in coalition formation games

In coalition formation games with arbitrary collection of permissible coalitions, a top/stable-coalition condition is sufficient for the existence of a unique core partition, strengthening uniqueness results both in Pycia (2012) and in Banerjee et al. (2001).

Long-term vegetation changes in four types of wetland in China and USA between 2000 and 2011: observations from MODIS

ABSTRACTWetland vegetation assessment is a critical component of evaluating wetland protective policies and providing useful information for global climate change research. In this study, four types of wetland are chosen to evaluate their long-term changes and potential linkage to anthropogenic (e. g., agricultural activities) and natural influences (e. g., hydrometeorological effects). These wetland types include the Yancheng wetland (coastal wetland), the Chongming wetland (river wetland), the Hongze Lake wetland (lake wetland) and the Everglades wetland (marsh wetland) in the contrasting environments of China and USA. The vegetation coverages of these wetlands were evaluated from 2000 to 2011 by using over 2,200 Moderate Resolution Imaging Spectroradiometer (MODIS) 250 m resolution images. Four vegetation indices (VIs) were compared to evaluate their effectiveness in assessing relative changes: the Normalized Difference Vegetation Index (NDVI), the Floating Algae Index (FAI), the Enhanced Vegetation Index (EVI), and the Visible Atmospherically Resistance Index (VARI). FAI performance was relatively insensitive in terms of both the statistical error and the aerosol effects compared with other VIs and was thus chosen to study the long-term vegetation changes. The results showed that 1) agricultural influence appeared to be relatively minimal compared with hydrometeorological effects in the Everglades wetland and in the core partitions of the Yancheng, Chongming, and Hongze Lake wetlands over the 12-year period, and 2) the entire partitions of the Yancheng, Chongming and Hongze Lake wetlands showed noticeable influences from agricultural activities in addition to natural variability. The cost-effective method that we demonstrate in this study may be extended to other wetlands near the same geographical location and to other satellite platforms.

Even [formula omitted]-core partitions and self-associate characters of [formula omitted

We extend a method of Olsson and Bessenrodt to determine the number of even partitions that are simultaneously s̄-core and t̄-core. When p and q are distinct primes, this also determines the number of self-associate characters of S̃n that are simultaneously defect 0 for p and q.

Simultaneous core multipartitions

We initiate the study of simultaneous core multipartitions, generalising simultaneous core partitions, which have been studied extensively in the recent literature. Given a multipartition datum(s∣c), which consists of a non-negative integer s and an l-tuple c of integers, we introduce the notion of an (s∣c)-core multipartition. Given an arbitrary set of multipartition data, we give necessary and sufficient conditions for the corresponding set of simultaneous core multipartitions to be finite. We then study the special case of simultaneous core bipartitions, giving exact enumerative results in some special subcases.