I have a dataset I want to use to generate a probability distribution. The distribution is skewed and can only include positive integers. I've tried normal (both skewed and truncated, although I can't figure out how to try both at once using R), poisson, and gamma. I've also plotted it on a Cullen and Frey graph (below). Gamma is by far the best fit of these. Is there an option with a better fit than a gamma distribution or way to find a better option? I'm concerned because the gamma distribution doesn't take into account the requirement that the data be positive integers. Thanks in advance.

These are my data (number of species per genus in Catarrhini, if that helps at all): c(16,5,2,1,4,2,1,1,1,1,19,5,5,8,1,10,2,4,1,1,7,3,1,2,2,2,1)

Edited for tags.