The conditional Monte Carlo technique advanced by Tukey et al. [1, 2] has been explained in analytic terms by Hammersley . This note offers an alternative explanation, wherein the group-theoretic aspect of the problem plays the dominant role. The method is illustrated on an example simpler than that treated in [1, 2].
"Groups and Conditional Monte Carlo." Ann. Math. Statist. 28 (4) 1048 - 1052, December, 1957. https://doi.org/10.1214/aoms/1177706812