On <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml9" display="inline" overflow="scroll" altimg="si9.gif"><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>-core partitions with distinct parts

Yan, Sherry H. F.; Qin, Guizhi; Jin, Zemin; Zhou, Robin D. P.

doi:10.1016/j.disc.2017.01.021

Cited by 14 publications

(18 citation statements)

References 14 publications

(19 reference statements)

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“…This conjecture has been proven by H.Xiong [14]: Theorem 1. (Xiong,15) For (a, a + 1)-core partitions with distinct parts, we have (1) the number of such partitions equals to the Fibonacci number F a+1 ;…”

Section: Introductionmentioning

confidence: 99%

Cores with distinct parts and bigraded Fibonacci numbers

Paramonov

2018

Discrete Mathematics

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The notion of (a, b)-cores is closely related to rational (a, b) Dyck paths due to Anderson's bijection, and thus the number of (a, a + 1)-cores is given by the Catalan number C a . Recent research shows that (a, a + 1) cores with distinct parts are enumerated by another important sequence-Fibonacci numbers F a . In this paper, we consider the abacus description of (a, b)-cores to introduce the natural grading and generalize this result to (a, as + 1)-cores. We also use the bijection with Dyck paths to count the number of (2k − 1, 2k + 1)-cores with distinct parts. We give a second grading to Fibonacci numbers, induced by bigraded Catalan sequence C a,b (q, t).Part (1) of the above theorem was independently proved by A.Straub [13]. Another interesting conjecture of Amdeberhan is the number of (2k − 1, 2k + 1)-cores with distinct parts. This conjecture have been proven by Yan, Qin, Jin and Zhou [15]:Theorem 2. (YQJZ,16) The number of (2k − 1, 2k + 1)-cores with distinct parts is equal to 2 2k−2 . The proof uses somewhat complicated arguments about the poset structure of cores. Results by Zaleski and Zeilberger [17] improve the argument using Experimental Mathematics tools in Maple. More recently Baek, Nam and Yu provided simpler bijective proof in [6]. 1 arXiv:1705.09991v1 [math.CO]

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Section: Introductionmentioning

confidence: 99%

Cores with distinct parts and bigraded Fibonacci numbers

Paramonov

2018

Discrete Mathematics

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“…We derive several polynomiality results and asymptotic formulas for moments of sizes of random (n, dn±1)-core partitions with distinct parts, which prove several conjectures of Zaleski [30]. In the past few years, the numbers, the largest sizes and the average sizes of (n, n+1), (2n+1, 2n+3)-core partitions with distinct parts were also well studied by many mathematicians (see [5,13,17,20,23,26,28,29]). But for general (s, t)-core partitions with distinct parts, even for the (n, n + 3)-core case, we know very little.…”

Section: Further Directionsmentioning

confidence: 65%

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

Xiong

Zang

2019

Sci. China Math.

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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] 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 E[X k n,n+1 ] of X n,n+1 is asymptotically equal to n 2 /10 k when 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 on the polynomiality of the expectation and higher moments of X n,dn−1 .

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“…(1) is conjectured by Amdeberhan [2] and proved in [22] and [25], while (2) is also conjectured by Amdeberhan [2] and proved in [27] and [29].…”

Section: Core Partitions With Distinct Partsmentioning

confidence: 77%

“…Core partitions of numerous types of additional restrictions have long been studied, since they are closely related to the representation of symmetric group [15], the theory of cranks [13], Dyck-paths [1,3,28], and Euler's theorem [22]. To solve core problems, mathematicians provide many different tools, including t-abacus [3,15], Hasse diagram [27,28] and even ideas from quantum mechanics [16].…”

Section: Introductionmentioning

confidence: 99%

Bijections between t-core partitions and t-tuples

Zhong

2020

Discrete Mathematics

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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.

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On (2k+1,2k+3)-core partitions with distinct parts

Abstract: Abstract. In this paper, we are mainly concerned with the enumeration of (2k + 1, 2k + 3)-core partitions with distinct parts. We derive the number and the largest size of such partitions, confirming two conjectures posed by Straub.

Cited by 14 publications

References 14 publications

Cores with distinct parts and bigraded Fibonacci numbers

Cores with distinct parts and bigraded Fibonacci numbers

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

Bijections between t-core partitions and t-tuples

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