data.table - How to count overlaps between multiple categories in R -

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i have data frame of start time, end time in unix time stamp , category, below:

enter image description here

i want find overlaps of 1 category "a" among other categories "b" & "c" , same other categories. want find how many number of categories overlapped how many times e.g. 2 categories overlapped 3 times , 3 categories overlapped 5 number of time. , average of duration overlapped shown below.

enter image description here


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