How to change the order and color scheme of stacked bar charts using ggplot2?

I am trying to get these ordered so that Space is stacked on top of Time and then order them in ascending order of Time. I also want to be able to pick the colors for each stack. Any help would be appreciated!Thanks a lot!

Data below:

structure(list(Beg = structure(c(20L, 19L, 18L, 15L, 1L, 3L, 
    6L, 10L, 13L, 8L, 5L, 11L, 9L, 7L, 2L, 4L, 17L, 16L, 14L, 12L, 
    20L, 19L, 18L, 15L, 1L, 3L, 6L, 10L, 13L, 8L, 5L, 11L, 9L, 7L, 
    2L, 4L, 17L, 16L, 14L, 12L), .Label = c("a", "b", "c", "d", "e", 
    "f", "g", "h", "i", "j", "k", "l", "m", "n", "o", "p", "q", "r", 
    "s", "t"), class = "factor"), Cat = structure(c(2L, 2L, 2L, 2L, 
    2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
    1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
    1L, 1L, 1L, 1L), .Label = c("Time", "Space"), class = "factor"), 
        Count = c(7824.92, 1006.79, 3570.93, 1484.5, 2885.32, 4194.84, 
        4348.94, 3603.31, 4826.33, 2225.49, 3350.02, 3778.35, 2698.51, 
        2247.01, 1705.17, 4742.72, 15231.15, 14083.26, 4437.68, 3109.09, 
        18875.45, 25816.95, 20836.93, 25501.53, 23996.55, 19427.12, 
        21467.89, 22472.71, 9876.27, 9548.99, 22171.83, 21179.33, 
        23358.26, 24763.62, 24551.94, 16726.11, 10691.68, 10537.26, 
        18012.88, 21453.15)), row.names = c(NA, -40L), class = "data.frame")

Solution

Adding to @NotThatKindODr's answer, you can order the bars in ascending order of time by reordering them with the fct_reorder function from the forcats package:

library(dplyr)
library(forcats)

df <- df %>% 
  mutate(Cat = fct_rev(Cat),
         Beg = fct_reorder(Beg, Count, max, .desc = T))

ggplot(df, aes(x = Beg, y = Count, fill = Cat)) + 
  geom_col() + 
  ggtitle("All Stuff") +
  theme_classic() + 
  coord_flip()

Which gives: