THE 2-ADIC VALUATIONS OF STIRLING NUMBERS OF THE SECOND KIND
نویسندگان
چکیده
منابع مشابه
Congruence Classes of 2-adic Valuations of Stirling Numbers of the Second Kind
We analyze congruence classes of S(n, k), the Stirling numbers of the second kind, modulo powers of 2. This analysis provides insight into a conjecture posed by Amdeberhan, Manna and Moll, which those authors established for k at most 5. We provide a framework that can be used to justify the conjecture by computational means, which we then complete for values of k between 5 and 20.
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In this paper, we investigate the 2-adic valuation of the Stirling numbers S(n, k) of the second kind. We show that v2(S(2n + 1, k + 1)) = s2(n) − 1 for any positive integer n, where s2(n) is the sum of binary digits of n. This confirms a conjecture of Amdeberhan, Manna and Moll. We show also that v2(S(4i, 5)) = v2(S(4i + 3, 5)) if and only if i 6≡ 7 (mod 32). This proves another conjecture of ...
متن کاملAlternative Proofs on the 2-adic Order of Stirling Numbers of the Second Kind
An interesting 2-adic property of the Stirling numbers of the second kind S(n, k) was conjectured by the author in 1994 and proved by De Wannemacker in 2005: ν2(S(2, k)) = d2(k) − 1, 1 ≤ k ≤ 2n. It was later generalized to ν2(S(c2, k)) = d2(k) − 1, 1 ≤ k ≤ 2n, c ≥ 1 by the author in 2009. Here we provide full and two partial alternative proofs of the generalized version. The proofs are based on...
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Let n and k be positive integers, d(k) and ν2(k) denote the number of ones in the binary representation of k and the highest power of two dividing k, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that ν2(S(2, k)) = d(k)−1, 1 ≤ k ≤ 2. Here we prove that ν2(S(c2, k)) = d(k)−1, 1 ≤ k ≤ 2, for any positive integer c. We improve and extend this statement in...
متن کاملOn 2-adic Orders of Stirling Numbers of the Second Kind
We prove that for any k = 1, . . . , 2 the 2-adic order of the Stirling number S(2, k) of the second kind is exactly d(k) − 1, where d(k) denotes the number of 1’s among the binary digits of k. This confirms a conjecture of Lengyel.
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ژورنال
عنوان ژورنال: International Journal of Number Theory
سال: 2012
ISSN: 1793-0421,1793-7310
DOI: 10.1142/s1793042112500625