R Combinatoric Sequence Generation - r

Question

I am attempting to generate a matrix where each column represents a sequence of factors in R. The factors can assume the values 1, 2, 3 or 4. Each sequence has 13 elements suggesting a total of 4^13 potential sequences. However, only a specific subset of these potential sequences are considered valid. The logic is as follows:
A sequence can start at any factor
If a sequence starts at 4, its second element can be less than or equal to 4
Once an element drops below 4, subsequent entries must be weakly increasing
If a sequence starts with 1, 2, or 3 it must be weakly increasing
So for example, the sequence (1,2,3,3,3.....3) is valid. The sequence (4,4,1,1,2,4,4....4) is also valid. The sequence (4,1,2,3,1,1....1) is not, since it is not weakly increasing after the first drop from 4 to 1.
At the moment, I have code to combine the 2&3 factors and generate this matrix. The process involves generating a matrix of all possible sequences and then filtering down based on the above logic. This is highly inefficient, but I can post it if necessary. This process also cannot be generalized to a four factor model, as the 4^13 potential sequences overwhelm my machine.
If any of you can offer insight into how I might generate these valid sequences, it would be greatly appreciated. Thank you.

Answer 1

I am assuming that once a gradually increasing vector reaches 4, it cannot jump back down again to a lower value the way that it can if 4 is the first number (if it can, the code is actually easier).
The following function generates compatible sequences, essentially using switch to implement a Markov chain
generate_seq <- function(n)
{
x <- numeric(n)
x[1] <- sample(4, 1)
had_a_four <- FALSE
for(i in seq(n - 1)) {
if(!had_a_four)
{
x[i + 1] <- switch(x[i], sample(1:2, 1, prob = c(3, 1)),
sample(2:3, 1, prob = c(3, 1)),
sample(3:4, 1, prob = c(3, 1)),
sample(4, 1))
}
else
{
x[i + 1] <- switch(x[i], sample(1:2, 1, prob = c(3, 1)),
sample(2:3, 1, prob = c(3, 1)),
sample(3:4, 1, prob = c(3, 1)),
4)
}
if(x[i + 1] == 4 & !all(x[1:(i+1)] == 4)) had_a_four <- TRUE
}
x
}
And we can generate a 4-column matrix like this:
set.seed(4)
matrix(replicate(4, generate_seq(13)), ncol = 4)
#> [,1] [,2] [,3] [,4]
#> [1,] 4 4 1 1
#> [2,] 3 3 1 2
#> [3,] 3 4 2 3
#> [4,] 3 4 2 3
#> [5,] 4 4 2 4
#> [6,] 4 4 3 4
#> [7,] 4 4 3 4
#> [8,] 4 4 3 4
#> [9,] 4 4 4 4
#> [10,] 4 4 4 4
#> [11,] 4 4 4 4
#> [12,] 4 4 4 4
#> [13,] 4 4 4 4

Answer 2

I think you can use RcppAlgos to do this efficiently by generating the combinations for vectors of length 1:n (where it's assumed that the shorter vectors are left padded to length 13 with 4) :
library(RcppAlgos)
get_combos <- function(n) {
unique(do.call(rbind, sapply(rev(seq(n)), function(x)
do.call(
cbind, c(rep(4, n - x), list(comboGeneral(1:4, x, TRUE)))
))))
}
res <- get_combos(13)
head(res)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
[1,] 1 1 1 1 1 1 1 1 1 1 1 1 1
[2,] 1 1 1 1 1 1 1 1 1 1 1 1 2
[3,] 1 1 1 1 1 1 1 1 1 1 1 1 3
[4,] 1 1 1 1 1 1 1 1 1 1 1 1 4
[5,] 1 1 1 1 1 1 1 1 1 1 1 2 2
[6,] 1 1 1 1 1 1 1 1 1 1 1 2 3
nrow(res)
[1] 2367