Using Kummer's Theorem, we give a necessary and sufficient condition for a
Narayana number to be divisible by a given prime. We use this to derive certain
properties of the Narayana triangle.Comment: 5 pages, see related papers at http://www.math.msu.edu/~saga

Bona, Miklos

Sagan, Bruce

English

arXiv

Abstract Using Kummer&apos;s Theorem, we give a necessary and sufficient condition for a Narayana number to be divisible by a given prime. We use this to derive certain properties of the Narayana triangle. 1 The main theorem Let N denote the nonnegative integers and let k, n 2 N. The Narayana numbers [10, A001263] can be defined as N (n, k)  = 1n `nk&apos; ` nk + 1&apos; where 0 &lt; = k &lt; n. The Narayana numbers (in fact, a q-analogue of them) were first studied by MacMahon [6, Article 495] and were later rediscovered by Narayana [7]. They are closely related to the Catalan numbers [10, A000108] Cn = 1n + 1 `2nn &apos; and in fact Pk N (n, k)  = Cn. The Narayana numbers can be arranged in a triangular array with N (n, k) in row n and column k so that the row sums are the Catalan numbers. Like the numbers Cn, the numbers N (n, k) have many combinatorial interpretations; see, for example, the article of Sulanke [11]. 

On divisibility of Narayana numbers by primes

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