library(combinat)
seated <- 10
perms <- matrix(unlist(permn(seated-1)), ncol = seated-1, byrow = TRUE)
permsextended <- cbind(1, perms+1, 1)
pairs <- 100 * permsextended[,-(seated+1)] + permsextended[,-1]
originalpairs <- c(100*(1:(seated-1)) + (2:seated), 100*seated + 1,
                   100*(2:seated) + (1:(seated-1)), 100*1 + seated)
dupes <- matrix(pairs %in% originalpairs, ncol=seated)
totaldupes <- rowSums(dupes)
nodupes <- permsextended[totaldupes==0, -(seated+1)]

> nrow(nodupes)
[1] 29926
> nrow(nodupes) / factorial(seated-1)
[1] 0.08246803

> head(nodupes)
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,]    1    7    2    9    3   10    4    6    8     5
[2,]    1    7    2   10    3    9    4    6    8     5
[3,]    1    7    2    9    3    8    4    6   10     5
[4,]    1    7    2   10    3    8    4    6    9     5
[5,]    1    7    2    8    3   10    4    6    9     5
[6,]    1    7    2    8    3    9    4    6   10     5

> table(nodupes[,6])
   2    3    4    5    6    7    8    9   10
4318 2844 3186 3048 3134 3048 3186 2844 4318

set.seed(1)
cases <- 1000000
seated <- 10 # should be less than 100
originalpairs <- c(100*(1:(seated-1)) + (2:seated), 100*seated + 1,
                   100*(2:seated) + (1:(seated-1)), 100*1 + seated)
runningcount <- 0
for (i in 1:cases){
    example <- sample(seated)
    examplextend <- c(example, example[1])
    examplepairs <- 100 * examplextend[-(seated+1)] + examplextend[-1]
    runningcount <- runningcount + (sum(examplepairs %in% originalpairs)==0)
    }

> runningcount / cases
[1] 0.082199