Abaci Structures of $(s, ms\pm1)$-Core Partitions

Nath, Rishi; Sellers, James A.

doi:10.37236/6476

Cited by 11 publications

(31 citation statements)

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“…Another set of combinatorial objects that caught the attention of a number of researchers [13,12,16] is the set of (a, as − 1)-cores with distinct parts. In particular, there is a Fibonacci-like recursive relation for the number of such cores: Theorem 3.…”

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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“…The (n, dn + 1)-core analog was obtained later by Nath-Sellers [14]. Theorem 1.4 (Nath-Sellers [14]).…”

Section: Introductionmentioning

confidence: 86%

“…The (n, dn + 1)-core analog was obtained later by Nath-Sellers [14]. Theorem 1.4 (Nath-Sellers [14]). Let M d (−1) = 0, M d (0) = 1, and M d (n) be the number of (n, dn + 1)-core partitions with distinct parts for two positive integers d and n. Then Table 1 gives the first few values for N d (n) and M d (n).…”

Section: Introductionmentioning

confidence: 86%

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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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“…Towards that problem, it was shown in [18] that (s, ms − 1)-core partitions into distinct parts are counted by Fibonacci-like numbers. This count was generalized by Nath and Sellers [13] to also include (s, ms + 1)-core partitions. Following [13], (s, ms ± 1)-core partitions refer to (s, ms − 1)-core or (s, ms + 1)-core partitions.…”

Section: Introductionmentioning

confidence: 99%

Refined Counting of Core Partitions into $d$-Distinct Parts

Burson

Sisneros-Thiry

Straub

2021

Electron. J. Combin.

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Using a combinatorial bijection with certain abaci diagrams, Nath and Sellers have enumerated $(s,ms\pm 1)$-core partitions into distinct parts. We generalize their result in several directions by including the number of parts of these partitions, by considering $d$-distinct partitions, and by allowing more general $(s,ms\pm r)$-core partitions. As an application of our approach, we obtain the average and maximum number of parts of these core partitions.

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Abaci Structures of $(s, ms\pm1)$-Core Partitions

Cited by 11 publications

References 11 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

Refined Counting of Core Partitions into $d$-Distinct Parts

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