The q-Dobinski formula may be interpreted as the average of powers of random variable X q with the q-Poisson distribution. Forty years ago Rota G. C. [1] proved the exponential generating function for Bell numbers B n to be of the form ∞ n=0 x n n! (B n) = exp(e x − 1) (1) using the linear functional L such that L(X n) = 1, n ≥ 0 (2) Then Bell numbers (see: formula (4) in [1]) are defined by L(...

‎Suppose G is a group of order p^2q^2 where p>q are prime numbers and suppose P and Q are Sylow p-subgroups and Sylow q-subgroups of G, ‎respectively‎. ‎In this paper‎, ‎we show that up to isomorphism‎, ‎there are four groups of order p^2q^2 when Q and P are cyclic‎, ‎three groups when Q is a cyclic and P is an elementary ablian group‎, ‎p^2+3p/2+7 groups when Q is an elementary ablian group an...

lexicographic ordering by spectral moments ($s$-order) among all trees is discussed in this‎ ‎paper‎. ‎for two given positive integers $p$ and $q$ with $pleqslant q$‎, ‎we denote $mathscr{t}_n^{p‎, ‎q}={t‎: ‎t$ is a tree of order $n$ with a $(p‎, ‎q)$-bipartition}‎. furthermore, ‎the last four trees‎, ‎in the $s$-order‎, ‎among $mathscr{t}_n^{p‎, ‎q},(4leqslant pleqslant q)$ are characterized‎.