• Bulletin of the Korean Mathematical Society
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• v.51 no.4
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• pp.1229-1240
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• 2014

In this paper we provide three results involving large Schr$\ddot{o}$der paths. First, we enumerate the number of large Schr$\ddot{o}$der paths by type. Second, we prove that these numbers are the coefficients of a certain symmetric function defined on the staircase skew shape when expanded in elementary symmetric functions. Finally we define a symmetric function on a Fuss path associated with its low valleys and prove that when expanded in elementary symmetric functions the indices are running over the types of all Schr$\ddot{o}$der paths. These results extend their counterparts of Kreweras and Armstrong-Eu on Dyck paths respectively.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.491-499
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• 2019

The problem of finding fast computing methods for Bernoulli numbers has a long and interesting history. In this paper, the author provides new proofs for two lacunary recurrence relations with gaps of length four and eight for the Bernoulli numbers. These proofs invoked the fact that the nth powers of ${\pi}^2$, ${\pi}^4$ and ${\pi}^8$ can be expressed in terms of the nth elementary symmetric functions.