Lowest unique common integer code

lupa.R

generate_fks <- function(N) {
  fki <- log(N + 1)
  fks <- c(fki)

  for (i in seq(N)) {
    fki <- log(exp(fki) - fki)
    fks <- c(fks, fki)
  }

  fks
}

normalise_vec <- function(v) {
  v / sum(v)
}


#' Probability that there is no one on k
mu <- function(k, P, N) {
  (1 - P[k]) ^ (N - 1)
}

#' Probability that there is no one unique BELOW j given that no one is on k
nu <- function(j, k, P, N) {
  if (j == 1) {
    return(1)
  }
  lhs <- nu(j-1, k, P, N)
  rhs <- 1 - ((N-1) * (P[j-1] / sum(P[-k])) * sum(P[-c(j-1, k)])^(N-2))
  lhs * rhs
}

#' EXPECTATION of picking i if a population of N-1 play according to P
E <- function(i, P, N) {
  mu(i, P, N) * nu(i, i, P, N)
}


N <- 100
P <- generate_fks(N) %>% normalise_vec

sapply(seq(N), E, P=P, N=N) %>% plot
sapply(seq(N), function(x) nu(x, x, P, N)) %>% plot

nu(50, 50, P, N)




j <- 50
k <- 50