Is there a pattern for this sequence?

Question

I've been looking at this and I thought there might be a pattern but I can't seem to find it. There are always $4$ elements, and they can be divided into $1, 2, 3, 4$ subsets ("chunks"). For example $$ S = \{ a, b, c, d \} $$ I want a function with the properties: \begin{align} f(1, S) &= \{ \{a, b, c, d \} \} \\ f(2, S) &= \{ \{ a \}, \{ b, c, d \} \} \\ f(3, S) &= \{ \{ a \}, \{ b, c \}, \{ d \} \\ f(4, S) &= \{ \{ a \}, \{ b \}, \{ c \}, \{ d \} \} \end{align}

I've been looking at it but I can't seem to find a rule that will produce the sequence of lengths. I'm looking for. Any ideas?

Answer 1

There are lots of imaginable principles that would lead to the partitions you show, but you don't have nearly enough data to choose in a principled way which of them is the once you need in your (undisclosed) application.

For example, if you had 10 elements instead of four, the first few patterns could be

abcdefghij
a,bcdefghij
a,bcdefghi,j
a,b,cdefghi,j
a,b,cdefgh,i,j
a,b,c,defgh,i,j

or

abcdefghij
a,bcdefghij
a,bc,defghij
a,bc,def,ghij
a,b,c,def,ghij
a,b,c,def,ghi,j

or even

abcdefghij
a,bcdefghij
a,bc,defghij
a,b,c,defghij
a,b,c,de,fghij
a,b,c,d,e,fghij

All of these generalize your 4-element division in a relatively straightforward way -- do you have any information that would allow you to judge which of these straightforward ways is better?