East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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ON THE TOUCHARD POLYNOMIALS AND MULTIPLICATIVE PLANE PARTITIONS

  • Kim, JunKyo (Department of Mathematics, Pusan National University)
  • Received : 2020.10.30
  • Accepted : 2020.11.08
  • Published : 2021.01.31
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Abstract

For a positive integer n, let μd(n) be the number of multiplicative d-dimensional partitions of ${\prod\limits_{i=1}^{n}}p_i$, where pi denotes the ith prime. The number of multiplicative partitions of a square free number with n prime factors is the Bell number μ1(n) = ��n. By the definition of the function μd(n), it can be seen that for all positive integers n, μ1(n) = Tn(1) = ��n, where Tn(x) is the nth Touchard (or exponential ) polynomial. We show that, for a positive n, μ2(n) = 2nTn(1/2). We also conjecture that for all m, μ3(m) ≤ 3mTm(1/3).

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