How can I plot the counts of my observed variable against the counts of a different level of the observed variable in a scatterplot?

How can I plot the counts of my observed variable against the counts of a
different level of the observed variable in a scatterplot?

I'd like to create scatter plots of the different conflict types against
each other. Each numbered sample would be a point. Samples should be
removed when there isn't aren't conflicts of the two types being plotted
against each other. I want to look at each relationship between conflict
types.
I made an example plot below of what I've been working on below but it
required lines and lines of explicit hard coding.

df <- structure(list(Name = c("487", "487", "232", "276", "405", "19",
"477", "487", "232", "159", "19", "232", "405", "487", "276",
"159", "159", "19", "286", "232", "232", "174", "19", "405",
"286", "599", "232", "30", "477", "232", "477", "477", "477",
"477", "477", "286", "159", "721", "487", "487", "276", "30",
"660", "345", "345", "345", "345", "174", "477", "355", "477",
"174", "159", "30", "477", "345", "276", "405", "608", "286",
"30", "232", "19", "721", "721", "102", "599", "345", "430",
"296", "174", "174", "174", "430", "430", "477", "487", "487",
"430", "430", "487", "345", "702", "30", "19", "30", "702", "702",
"174", "477", "174", "355", "159", "232", "430", "296", "702",
"286", "430", "702", "702", "712", "345", "184", "721", "184",
"102", "296", "599", "702", "702", "702", "38", "38", "38", "430",
"430", "430", "477", "477", "430", "477", "660", "702", "702",
"660", "660", "660", "702", "296", "702", "174", "296", "296",
"345", "702", "660", "599", "417", "417", "417", "702", "652",
"345", "184", "591", "184", "184", "599", "417", "90", "599",
"430", "678", "678", "38", "38", "477", "678", "678", "477",
"678", "678", "417", "430", "430", "572", "572", "572", "702",
"572", "702", "702", "702", "442", "442", "678", "442", "296",
"30", "311", "311", "311", "311", "311", "311", "591", "591",
"102", "442", "608", "430", "194", "194", "194", "678", "678",
"678", "38", "38", "678", "38", "38", "49", "49", "49", "690",
"690", "690", "572", "690", "296", "296", "552", "690", "702",
"580", "580", "580", "580", "454", "454", "454", "702", "702",
"345", "311", "311", "311", "184", "608", "194", "102", "224",
"224", "194", "194", "194", "660", "660", "38", "660", "660",
"90", "90", "212", "212", "212", "212", "690", "668", "690",
"690", "442", "562", "562", "562", "562", "562", "562", "562",
"562", "417", "417", "652", "652", "320", "320", "320", "184",
"599", "430", "678", "678", "62", "62", "62", "62", "62", "62",
"62", "62", "62", "62", "62", "62", "62", "62", "62", "62", "62",
"62", "62", "690", "62", "62", "62", "62", "62", "62", "62",
"62", "62", "62", "62", "62", "62", "62", "62", "591", "194",
"62", "62", "62", "62", "62", "62", "62", "62", "62", "62", "62",
"62", "62", "62", "62", "62", "62", "62", "62", "62", "62", "62",
"62", "62", "62", "62", "62", "62", "62", "62", "62", "62", "62",
"62", "62", "62", "62", "62", "62", "62", "102", "102", "224",
"329", "329", "224", "224", "224", "224", "224", "660", "212",
"212", "212", "212", "203", "203", "203", "203", "212", "212",
"668", "62", "76", "76", "76", "76", "76", "76", "76", "76",
"76", "76", "76", "76", "76", "652", "652", "62", "62", "430",
"678"), Conflict = structure(c(2L, 2L, 2L, 1L, 1L, 1L, 2L, 3L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 3L, 2L, 1L, 4L, 1L, 1L,
2L, 2L, 1L, 1L, 2L, 2L, 2L, 3L, 4L, 4L, 1L, 1L, 1L, 2L, 2L, 2L,
1L, 3L, 2L, 2L, 1L, 2L, 2L, 1L, 2L, 2L, 1L, 3L, 2L, 2L, 2L, 2L,
2L, 4L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 1L, 1L, 2L,
1L, 1L, 3L, 2L, 2L, 2L, 1L, 2L, 2L, 2L, 3L, 2L, 2L, 2L, 2L, 2L,
2L, 4L, 3L, 2L, 1L, 2L, 1L, 1L, 2L, 1L, 1L, 2L, 2L, 1L, 3L, 3L,
4L, 1L, 2L, 2L, 2L, 2L, 2L, 3L, 2L, 1L, 2L, 2L, 4L, 1L, 2L, 3L,
1L, 2L, 4L, 2L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 4L,
1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 4L, 1L, 1L, 2L, 2L, 1L, 2L,
1L, 3L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 1L, 3L, 3L, 1L, 3L, 2L, 2L,
3L, 2L, 2L, 2L, 1L, 2L, 1L, 2L, 2L, 1L, 2L, 2L, 1L, 2L, 1L, 2L,
2L, 1L, 1L, 2L, 2L, 1L, 2L, 1L, 4L, 2L, 2L, 1L, 2L, 2L, 2L, 2L,
2L, 1L, 1L, 2L, 2L, 2L, 3L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 2L,
2L, 2L, 3L, 2L, 3L, 2L, 2L, 2L, 3L, 1L, 1L, 1L, 2L, 2L, 3L, 2L,
1L, 1L, 3L, 3L, 1L, 1L, 2L, 4L, 2L, 4L, 2L, 3L, 1L, 1L, 1L, 2L,
3L, 1L, 2L, 1L, 3L, 1L, 2L, 2L, 2L, 1L, 2L, 2L, 1L, 3L, 3L, 1L,
2L, 1L, 3L, 1L, 2L, 2L, 2L, 1L, 1L, 3L, 2L, 2L, 2L, 2L, 3L, 2L,
4L, 3L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L,
2L, 3L, 3L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 2L, 2L, 2L, 2L, 3L, 2L,
2L, 2L, 2L, 3L, 2L, 3L, 2L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 2L,
2L, 3L, 3L, 2L, 2L, 3L, 2L, 2L, 2L, 3L, 3L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 1L, 2L, 2L, 3L, 1L, 2L, 2L,
1L, 3L, 4L, 2L, 4L, 2L, 1L, 2L, 1L, 4L, 1L, 1L, 1L, 2L, 2L, 4L,
2L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L, 2L, 3L, 1L, 2L, 3L, 2L,
2L), .Label = c("DB Zero", "File Zero", "Het - Homo", "Homo - Het"
), class = "factor")), .Names = c("Name", "Conflict"), row.names = c(NA,
-393L), class = "data.frame")