# How do I find the number of unique permutations for a PERMANOVA with an unbalanced design?

From the PRIMER manual (p. 28 about Monte Carlo permutations; http://updates.primer-e.com/primer7/manuals/PERMANOVA+_manual.pdf), the number of unique permutations for a PERMANOVA with $$a$$ groups and $$n$$ replicates per group is: $$(an)![a!(n!)^a].$$ However, what is the calculation if the study design is unbalanced? For example, one group with $$4$$ individuals, another with $$5$$, another with $$6$$? Is there a way to calculate the number of unique permutations with adonis2?

Thank you for pointing numPerms(). However, I am getting an incorrect answer, even with simpler examples.

# Create 2 groups of 5.
myVec = c(rep("A",5), rep("B", 5))

# Define the permutation scheme.
hh = how(within = Within(type = "free"),
blocks = as.factor(myVec))

# Calculate number of permutations
numPerms(nobs(myVec), hh)


However, 2 groups of 5 should have 945 unique permutations, but this returns 14400. What did I do wrong?

• This is a special case of the general answer I gave at stats.stackexchange.com/a/415878/919.
– whuber
Jan 5, 2023 at 21:25
• Thank you for pointing me in the right direction, but I'm not sure if I know how to solve it for this case. Following your equations, what would the values be for k? Jan 5, 2023 at 22:44

Function permute::numPerms calculates the number of permutations for your current model. The permutations in vegan::adonis2 are based on this very same permute package, and if you have specific problems with permutations, you should consult the permute package. Use numPerms() and read its help page.