On the Touchard polynomials and Multiplicative Plane Partitions
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For a positive integer $n$, let $\MP_d(n)$ be the number of multiplicative $d$-dimensional partitions of $\prod\limits_{i = 1}^n p_i $, where $p_i$ denotes the $i$th prime. The number of multiplicative partitions of a square free number with $n$ prime factors is the Bell number $\MP_1(n)=\Bn$. By the definition of the function $\MP_d(n)$, it can be seen that for all positive integers $n$, $\MP_1(n)=T_n(1)=\Bn$, where $T_n(x)$ is the $n$th Touchard (or exponential ) polynomial. We show that, for a positive $n$, $\PL(n)= 2^n T_n(1/2)$. We also conjecture that for all $m$, $\SL(m) \le 3^m T_m(1/3)$.
Abstract
For a positive integer $n$, let $\MP_d(n)$ be the number of multiplicative $d$-dimensional partitions of $\prod\limits_{i = 1}^n p_i $, where $p_i$ denotes the $i$th prime. The number of multiplicative partitions of a square free number with $n$ prime factors is the Bell number $\MP_1(n)=\Bn$. By the definition of the function $\MP_d(n)$, it can be seen that for all positive integers $n$, $\MP_1(n)=T_n(1)=\Bn$, where $T_n(x)$ is the $n$th Touchard (or exponential ) polynomial. We show that, for a positive $n$, $\PL(n)= 2^n T_n(1/2)$. We also conjecture that for all $m$, $\SL(m) \le 3^m T_m(1/3)$.
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