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Consider a standard ${\Lambda }$-coalescent that comes down from infinity.Such a coalescent starts from a configuration consisting of infinitely manyblocks at time $0$, but its number of blocks $N_t$ is a finite random variableat each positive time $t$. Berestycki et al. [Ann. Probab. 38 (2010) 207-233]found the first-order approximation $v$ for the process $N$ at small times.This is a deterministic function satisfying $N_t/v_t\to1$ as $t\to0$. Thepresent paper reports on the first progress in the study of the second-orderasymptotics for $N$ at small times. We show that, if the driving measure$\Lambda$ has a density near zero which behaves as $x^{-\beta}$ with$\beta\in(0,1)$, then the process $(\varepsilon^{-1/(1+\beta)}(N_{\varepsilont}/v_{\varepsilon t}-1))_{t\ge0}$ converges in law as $\varepsilon\to0$ in theSkorokhod space to a totally skewed $(1+\beta)$-stable process. Moreover, thisprocess is a unique solution of a related stochastic differential equation ofOrnstein-Uhlenbeck type, with a completely asymmetric stable L\'{e}vy noise.

Second-order asymptotics for the block counting process in a class of regularly varying $\Lambda$-coalescents