R/ESF.R

In jack: Jack, Zonal, and Schur Polynomials

#' Evaluation of elementary symmetric functions
#'
#' Evaluates an elementary symmetric function.
#'
#' @param x a numeric vector or a \code{\link[gmp]{bigq}} vector
#' @param lambda an integer partition, given as a vector of decreasing
#' integers
#'
#' @return A number if \code{x} is numeric, a \code{bigq} rational number
#' if \code{x} is a \code{bigq} vector.
#' @importFrom gmp as.bigq is.bigq
#' @importFrom DescTools Permn
#' @export
#'
#' @examples x <- c(1, 2, 5/2)
#' lambda <- c(3, 1)
#' ESF(x, lambda)
#' library(gmp)
#' x <- c(as.bigq(1), as.bigq(2), as.bigq(5,2))
#' ESF(x, lambda)
ESF <- function(x, lambda){
  stopifnot(isPartition(lambda))
  gmp <- is.bigq(x)
  m <- length(x)
  lambda <- lambda[lambda>0]
  if(any(lambda > m)) return(if(gmp) as.bigq(0L) else 0)
  if(gmp){
    eks <- as.bigq(rep(0L, length(lambda)))
  }else{
    eks <- numeric(length(lambda))
  }
  for(k in seq_along(lambda)){
    kappa <- numeric(m)
    kappa[seq_len(lambda[k])] <- rep(1L, lambda[k])
    perms <- DescTools::Permn(kappa)
    if(gmp){
      ek <- as.bigq(0L)
      for(i in 1L:nrow(perms)){
        pows <- as.bigq(rep(0L, m))
        for(j in 1L:m){
          pows[j] <- x[j]^perms[i,j]
        }
        ek <- ek + prod(pows)
      }
    }else{
      ek <- 0
      for(i in 1L:nrow(perms)){
        ek <- ek + prod(x^perms[i,])
      }
    }
    eks[k] <- ek
  }
  prod(eks)
}