2017
DOI: 10.1016/j.disc.2017.01.021
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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.

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Cited by 14 publications
(18 citation 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%
“…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%
“…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%
“…(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%