Partitions into Distinct Parts and Elliptic Curves

Ono, Ken

doi:10.1006/jcta.1997.2847

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2018

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“…Lovejoy [17] studied the arithmetic properties of partitions with distinct parts. Ono [22] gave weighted recurrence relations for the number of partitions of n with distinct parts.…”

Section: Introductionmentioning

confidence: 99%

The Corners of Core Partitions

Huang¹,

Wang²

2018

SIAM J. Discrete Math.

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In this paper, we are concerned with counting corners of core partitions. We introduce the concepts of stitches and anti-stitches, which are pairs of cells in a quotient space which we call wrap-up space. We prove that the anti-stitches of a rational Dyck path are in bijection with the segments of structure sets of the corresponding core partition, therefore the corners of a core partition can be counted by the number of stitches or anti-stitches. Based on these results, for coprime positive integers a and b, we give two essentially different formulae for the number of corners in all (a, b)-cores. This leads to an unexpected identity, expressing the rational Catalan numbers as weighed sums of binomial numbers. Moreover, we show that for an (n, n + 1)-core partition λ determined by a certain (n, n + 1)-Dyck path P , the corners of λ correspond to pairs of consecutive right steps in P. As a consequence, we show that the number of (n, n + 1)-cores with k corners is the Narayana number N (n, k + 1). We also extend these results to multi-cores.

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“…Lovejoy [17] studied the arithmetic properties of partitions with distinct parts. Ono [22] gave weighted recurrence relations for the number of partitions of n with distinct parts.…”

Section: Introductionmentioning

confidence: 99%

The Corners of Core Partitions

Huang¹,

Wang²

2018

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

Partitions into Distinct Parts and Elliptic Curves

Cited by 1 publication

References 6 publications

The Corners of Core Partitions

The Corners of Core Partitions

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