Let [ n { = {1, …, n } and let Ω n be the set of all mappings from [ n { to itself. Let f be a random uniform element of Ω n and let T( f ) and B( f ) denote, respectively, the least common multiple and the product of the length of the cycles of f . Harris proved in 1973 that T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for random uniform mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k ( r ) ≥ 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x k + a over the integers modulo p , with p ≡ 1 (mod k). We exhibit and discuss our numerical results on this heuristic.

Let f be a uniformly random element of the set of all mappings from [n] = {1, ..., n} to itself. Let T(f) and B(f) denote, respectively, the least common multiple and the product of the lengths of the cycles of f. Harris proved in 1973 that log T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for uniform random mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k(r) >= 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x^k + a over the integers modulo p, where k divides p - 1. We exhibit and discuss our numerical results on this heuristic.

For random partitions of $[ n ]$, let $L_n $ and $R_n $, respectively, denote the maximum block size and its multiplicity. The average multiplicity is $E( R_n ) = H( \{ m_n \} ) + o( 1 )$ as $n \to \infty $, where H is an explicitly given analytic function and $\{ m_n \}$ is the fractional part of a certain implicitly defined root. The cumulative distribution function of $L_n $ also depends on $\{ m_n \}$. The sequence $\langle \{ m_n\}\rangle^{\infty}_{n=1}$ is dense in (0, 1). This establishes both the nonexistence of a limit distribution for $L_n $ and the nonexistence of a limiting value for $E( R_n )$.