The Pitman sampling formula has been extensively studied as a model for random partitions. One object of interest is the length \(\varvec{K}_{\varvec{n},\varvec{\theta },\varvec{\alpha }}\) of a random partition governed by this formula, where \(\varvec{n}\varvec{\in }\mathbb {N}\) , \(\varvec{\alpha }\varvec{\in }\varvec{(0,1)}\) , and \(\varvec{\theta } \varvec{\in } (-\varvec{\alpha },\varvec{\infty })\) are parameters. This paper investigates the asymptotic behavior of its r-th moment \(\mathsf{E}[\varvec{K}_{\varvec{n},\varvec{\theta },\varvec{\alpha }}^{\varvec{r}}]\) for \(\varvec{r} \varvec{\in } \{\varvec{1,2},\varvec{\ldots }\}\) under two distinct asymptotic regimes with \(\varvec{\alpha }\) fixed. First, we refine existing approximations of \(\mathsf{E}[\varvec{K}_{\varvec{n},\varvec{\theta },\varvec{\alpha }}^{\varvec{r}}]\) as \(\varvec{n}\varvec{\rightarrow }\varvec{\infty }\) , offering improved precision. Second, we derive new asymptotic evaluations when both \(\varvec{n}\) and \(\varvec{\theta }\) tend to infinity with \(\varvec{\theta }/\varvec{n} \varvec{\rightarrow } \varvec{0}\) . These results contribute to a deeper understanding of the asymptotic behavior of \(\varvec{K}_{\varvec{n},\varvec{\theta },\varvec{\alpha }}\) .
