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

Core partitions with distinct parts

“…Zaleski [23] obtained the explicit expressions for the moments of the sizes of (t, t + 1)-core partitions with distinct parts, which gave a generalization of Theorem 1.2 (4). Furthermore, Zaleski and Zeilberger [22] also obtained the the moments of the sizes of (2t + 1, 2t + 3)-core partitions with distinct parts, whose number, largest size and average size were given by Yan, Qin, Jin and Zhou [20]. Our main results are stated next.…”

“…Much attention has been attracted to simultaneous core partitions with distinct parts since Amdeberhan's conjectures [1] on this subject in 2015. The results on the enumeration of (t, t + 1), (t, t + 2) and (t, nt ± 1)-core partitions with distinct parts can be found in several papers [10,17,20,22,23] published in 2016 and 2017. In this paper 1 , we obtain the generating function of t-core partitions with distinct parts in Theorem 1.1.…”

Bijections between t-core partitions and t-tuples

“…The followings are some known fascinating results. (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].…”

“…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], 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].…”

“…They also proved that [λ] is a t-core if and only if there is no dot whose hook length is divisible by t.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], 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].This paper includes new proofs of some known theorems, which lead to several new results on core partitions. By giving different expressions for partitions, we aim to evaluate the cardinality of various types of core partitions.Section 2 reviews three bijections relating t-cores and t-tuples, and also includes their applications.…”